Nonlinear Differential Equations, Electronic Journal of Differential Equations, Conf. 05, 2000, pp 135–171. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

LARGE SOLUTIONS, METASOLUTIONS, AND

ASYMPTOTIC BEHAVIOUR OF THE REGULAR POSITIVE

SOLUTIONS OF SUBLINEAR PARABOLIC PROBLEMS

Julián López-Gómez

Dedicated to Alan C. Lazer on his 60th birthday

Abstract. In this paper we analyze the existence of regular and large positive solutions for a class of non-linear elliptic boundary value problems of logistic type in the presence of refuges. These solutions describe the asymptotic behaviour of the regular positive solutions of the associated parabolic model. The main tool in our analysis is an extension of the interior estimates found by J. B. Keller in [Ke57] and R. Osserman in [Os57] to cover the case of changing sign nonlinearities combined with the construction of adequate sub and supersolutions. The supersolutions are far from obvious since the nonlinearity vanishes in finitely many regions of the underlying support domain.

1. Introduction

In this paper we analyze the asymptotic behaviour of the positive solutions of

\frac{∂u}{∂t} − Δu = λu − a(x)f(x, u)u in Ω × (0, ∞) ,

u(x, t) = 0 on ∂Ω × (0, ∞) ,
u(·, 0) = u_{0} ≥ 0 in Ω ,

(1.1)

where Ω is a bounded domain of ℝ^{N}, N ≥ 1, with boundary ∂Ω of class C^{3}, λ ∈ ℝ, and a ≥ 0, a ≠ 0, is a function of class C^{μ}(Ω̄), for some μ ∈ (0, 1), satisfying the following assumptions: (Ha1) The open set

Ω_{+} := { x ∈ Ω : a(x) > 0 }

is connected with boundary ∂Ω_{+} of class C^{3}. Moreover, if Γ_{+} and Γ are components of ∂Ω_{+} and ∂Ω, respectively, such that Γ_{+}∩Γ ≠ ∅, then Γ_{+} = Γ and a(x) is bounded away from zero on Γ_{+}.

(Ha2) If Γ_{+} is a component of ∂Ω_{+} such that Γ_{+} ∩ ∂Ω = ∅, then

a(x) = o(dist(x, Γ_{+})) as dist(x, Γ_{+}) ↓ 0 , x ∈ Ω_{+} .  (1.2)

Mathematics Subject Classifications: 35K57, 35K60, 35D05. Key words: Large solutions. Metasolutions. Asymptotic behavior. \begin{matrix}c \\ ◯ \end{matrix}2000 Southwest Texas State University. Published October 25, 2000. Supported by the Spanish Grant DGES PB96-0621.

(Ha3) The open set Ω_{0} := Ω \ Ω̄_{+} possesses a finite number of components, say Ω_{0,j}^{i}, 1 ≤ i ≤ m, 1 ≤ j ≤ n_{i}, such that

Ω̄_{0,j}^{i} ∩ Ω̄î ; 0,ĵ = ∅ if (i, j) ≠ (î, ĵ) .

In the sequel given a regular subdomain D of Ω and V ∈ C^{μ}(D̄), we denote by σ^{D}[−Δ+V] the principal eigenvalue of −Δ+V in D under homogeneous Dirichlet boundary conditions, and set σ^{D} := σ^{D}[−Δ]. Without loss of generality we can label the components of Ω_{0} so that

σ^{Ω_{0,j}^{i}} = σ^{Ω_{0,j+1}^{i}} , 1 ≤ i ≤ m , 1 ≤ j ≤ n_{i} − 1 ,  (1.3a)

and

σ^{Ω_{0,1}^{i}} < σ^{Ω_{0,1}^{i+1}} , 1 ≤ i ≤ m − 1 .  (1.3b)

As for the nonlinearity, we suppose the following:

(Hf) The function f : Ω̄ × [0, ∞) → ℝ is continuous and of class C^{μ,1+μ}(Ω̄ × (0, ∞)) and it satisfies f(x, 0) = 0, f(x, u) > 0, and ∂_{u}f(x, u) > 0 for all u > 0 and x ∈ Ω. Moreover,

lim_{u↑∞} f(x, u) = ∞

uniformly on compact subsets of Ω̄_{+}.

Problem (1.1) provides us with the evolution of a single species obeying a generalized logistic growth law [Mu93], [Ok80]. Typically, u is the density of the species, Ω is the inhabiting region, λ is the net growth rate of u, and the coefficient a(x) measures the saturation effect responses to the population stress in Ω_{+}. In Ω_{0} the individuals of the population are free from other effects than diffusion and so each of the components of Ω_{0} can be regarded as a refuge. The refuges have been ordered accordingly to the size of the principal eigenvalue of −Δ under homogeneous Dirichlet boundary conditions. The refuges Ω_{0,j}^{1}, 1 ≤ j ≤ n_{i}, will be called the lower order refuges. Throughout this paper we use the following notation

σ_{0} := σ^{Ω} , σ_{i} := σ^{Ω_{0,1}^{i}} , 1 ≤ i ≤ m .  (1.4)

Thanks to (1.3b),

σ_{0} < σ_{1} < σ_{2} < ⋯ < σ_{m} .

In case m = 1 one should simply write σ_{0} < σ_{1}.

The classical results for the case when a(x) is bounded away from zero in Ω̄ strongly suggest that the dynamics of the positive solutions of (1.1) should be regulated by its non-negative steady states, which are the non-negative solutions of

−Δu = λu − a(x)f(x, u)u in Ω , u|_{∂Ω} = 0 .  (1.5)

Under the assumptions above, any non-negative solution of (1.5) lies in the Banach space

U := { u ∈ C^{2+μ}(Ω̄) : u|_{∂Ω} = 0 } .

Moreover, if u ∈ U \ {0} is a non-negative solution of (1.5), then it follows from the strong maximum principle that u(x) > 0 for all x ∈ Ω and \frac{∂u}{∂n}(x) < 0 for all x ∈ ∂Ω, where n is the outward unit normal to Ω at x, i.e. u lies in the interior of the cone U^{+} of non-negative functions of U. Furthermore, for any p > \frac{N}{2} and u_{0} ∈ U_{0} := W_{0}^{2,p}(Ω), u_{0} ≥ 0, the parabolic problem (1.1) has a unique global regular solution u_{[λ,a,Ω]}(x, t; u_{0}) (cf. Proposition 24.9 of [DK92]). By a global regular solution we mean that

u_{[λ,a,Ω]} ∈ C(Ω̄ × [0, ∞)) ∩ C^{2+μ,1+ \frac{μ}{2}} (Ω̄ × (0, ∞)) .  (1.6)

Indeed, for any non-negative solution u of (1.1) we have

\frac{∂u}{∂t} − Δu = λu − a(x)f(x, u)u ≤ λu ,

and hence

u(x, t; u_{0}) ≤ T (t)u_{0} ,

where T (t) is the L_{p}-evolution operator associated with Δ + λ under homogeneous Dirichlet boundary conditions. Therefore, the solutions are global in time. If the initial data has less regularity, e.g. u_{0} ∈ C(Ω̄) instead of u_{0} ∈ U_{0}, then (1.6) might fail. Nevertheless, by parabolic regularity, for any u_{0} ∈ C(Ω̄), u_{0} ≥ 0, we have

u_{[λ,a,Ω]} ∈ C^{2+μ,1+ \frac{μ}{2}} (Ω̄ × (0, ∞)) .

Some pioneer results about the dynamics of the positive solutions of (1.1) in the presence of refuges were found in [FKLM96], where assuming m = 1 and n_{1} = 1 it was shown that problem (1.5) possesses a positive solution if, and only if, σ_{0} < λ < σ_{1} and that within this range of values of the parameter λ the unique positive solution of (1.5) is a global attractor for the positive solutions of (1.1), whereas the species is driven to extinction if λ ≤ σ_{0}. Some pioneer results about the existence of positive solutions of (1.5) had already been given in [BO86] and [Ou92].

In [GGLS98] problem (1.5) was analyzed under more restrictive assumptions than those imposed here in. Precisely, it was assumed that m = n_{1} = 1, that D := Ω \ Ω̄_{0,1}^{1} satisfies D̄ ⊂ Ω, and that there exists p > 0 such that

f(x, u) = |u|^{p} , (x, u) ∈ Ω̄ × ℝ .  (1.7)

One of the many results found in [GGLS98] shows that if u_{λ} stands for the maximal regular non-negative steady-state of (1.5), then

lim_{λ↑σ_{1}} u_{λ} = ∞ uniformly in Ω̄_{0,1}^{1} ,

while in D := Ω \ Ω̄_{0,1}^{1} the positive solutions u_{λ} stabilize as λ ↑ σ_{1} to a large regular positive solution of the problem

\begin{matrix}−Δu = λu − a|u|^{p}u in D , \\ u = ∞ on ∂D , \end{matrix}  (1.8)

for the value of the parameter λ = σ_{1}. This result is of great interest by itself because it implies the existence of a large solution in a problem where the weight function a(x) vanishes on a component of the boundary, whereas most of the previous results about the existence of large solutions had been given in the simplest case when the nonlinearity is bounded away from zero (cf. [Ke57], [Os57], [BM91], [Ve92], [LM93], [LM94], [MV97], and the references therein), and in addition it shows how a uniform interior Harnack inequality might fail when the nonlinearity vanishes in some region of the support domain. It should be pointed out that the classical interior estimates of [Ke57] and [Os57] can not be applied straight away to show the existence of a large positive solution of (1.8), since our nonlinearities change of sign, and that this fact provokes the existence of some unimportant gaps in some of the proofs given in references (e.g. the proof of Lemma 1.3 in [MV97]). Nevertheless, the results of [LM93], [LM94] and [GGLS98] can be used to fill in these gaps, since the large solutions in balls provide us with those interior estimates.

Thanks to these results a rather natural question arises. How does behave the population as times grows to infinity when λ ≥ σ_{1}? It can not approach to a regular positive solution, of course. In fact, for any u_{0} > 0 and ε > 0 we have that

u_{[λ,a,Ω]}(·, t; u_{0}) ≥ u_{[σ_{1}−ε,a,Ω]}(·, t; u_{0}) ,

and hence

lim inf_{t↑∞} u_{[λ,a,Ω]}(·, t; u_{0}) ≥ u_{σ_{1}−ε} .

On the other hand, thanks to Theorem 3.1 of [GGLS98],

lim_{ε↓0} u_{σ_{1}−ε} = ∞ uniformly in Ω̄_{0,1}^{1} .

Therefore, for each λ ≥ σ_{1} it follows that

lim inf_{t↑∞} u_{[λ,a,Ω]}(·, t; u_{0}) = ∞ uniformly in Ω̄_{0,1}^{1} ,  (1.9)

and hence the population grows arbitrarily in the refuge as times passes by. How does behave the population in the complement of the refuge? The answer to this question in the very special case when m = 1, n_{1} = 1, Ω̄_{0,1}^{1} ⊂ Ω and (1.7) are satisfied was given very recently in [DH99], where it was shown that the limiting population in D as t ↑ ∞ lies in between the minimal and the maximal large solution of the following problem

−Δu = λu − a(x)u^{p+1} in D ,
u = ∞ on ∂D ∩ Ω ,
u = 0 on ∂D ∩ ∂Ω .

(1.10)

For this special location of the refuge one can consider a general boundary operator on ∂D ∩ ∂Ω of mixed tipe (cf. [FKLM96] and [DH99]). Note that in (1.10) a(x) is bounded away from zero in any compact subset of D.

The case when the species possesses an arbitrarily large number of refuges within its inhabiting region and (1.7) is satisfied has been already analyzed in [GL98] and [G99], where the concept of metasolution was introduced in order to characterize all the possible limiting profiles of the population as t ↑ ∞. Setting

Ω_{k} := Ω \ ∪_{i=1}^{k} ∪_{j=1}^{n_{i}} Ω̄_{0,j}^{i} , 1 ≤ k ≤ m ,

a function

M : Ω → [0, ∞]

is said to be a metasolution of order k of (1.5) supported in Ω_{k} if M|_{Ω_{k}} is a large regular solution of

−Δu = λu − a(x)uf(x, u) in Ω_{k} ,
u = ∞ on ∂Ω_{k} \ ∂Ω ,
u = 0 on ∂Ω_{k} ∩ ∂Ω ,

(1.11)

and

M = ∞ in (Ω \ Ω_{k}) ∪ (∂Ω_{k} \ ∂Ω) .

In other words, a metasolution is the continuous extension by infinity to the totality of Ω of a large solution of (1.11). The main analytical result obtained in [GL98] shows that (1.5) possesses a metasolution of order k supported in Ω_{k} provided

σ_{k} ≤ λ < σ_{k+1}

if k ≤ m − 1, and provided λ ≥ σ_{m} if k = m. This result was obtained for the special choice (1.7), where the interior estimates of [GGLS98] work out.

In this paper we obtain some substantial generalizations of all these results to cover the general case when conditions (Ha1-3) and (Hf) are satisfied, and in particular we complete the analysis already begun in [GL98] where it was conjectured that the metasolutions of order k ≤ m − 1 (resp k = m) should give the limiting profile of the population as t ↑ ∞ if σ_{k} ≤ λ < σ_{k+1} (resp. λ ≥ σ_{m}).

The distribution of this paper is as follows. In Section 2 we characterize the existence of regular positive solutions for (1.5) and show their global asymptotic stability with respect to the positive solutions of (1.1).

In Section 3 we generalize the interior estimates of J. B. Keller [Ke57] and R. Osserman [Os57] to cover the case when the nonlinearity changes of sign. Some previous results for changing sign nonlinearities were found by A. C. Lazer and P. J. McKenna in [LM94]. In the special case when a, λ ∈ ℝ and the nonlinearity has the form

h(u) := auf(u) − λu  (1.12)

it was assumed in [LM94] that h ∈ C^{1}([f^{−1}(\frac{λ}{a}), ∞) with h′ ≥ 0 and that in addition h′(u) is nondecreasing for u large, and

lim inf_{u↑∞} \frac{h′(u)}{\sqrt{∫_{f^{−1}( \frac{λ}{a})}^{u} h(z) dz}} > 0 .  (1.13)

Instead of these conditions, for the special choice (1.12) we only need to assume that there exists u_{∗} > f^{−1}(\frac{λ}{a}) such that

∫_{u_{∗}}^{∞} [∫_{u_{∗}}^{u} h(z) dz]^{− \frac{1}{2}} du < ∞ .  (1.14)

Note that for the choice (1.7), (1.14) is satisfied for any p > 0, while (1.13) is only satisfied for p ≥ 2. It should be pointed out that if f(u) satisfies (Hf), then

∫_{f^{−1}( \frac{λ}{a})}^{∞} [∫_{f^{−1}( \frac{λ}{a})}^{u} h(z) dz]^{− \frac{1}{2}} du = ∞ ,

and hence condition (2) of [Ke57] fails. Therefore, the interior estimates of [Ke57] can not be applied straight away to our problem.

Once obtained these interior estimates, we shall use them to get some very general results about the existence of large solutions going to infinity on some of the components of the boundary where a(x) is bounded away from zero. These results are completely new, since a(x) can vanish on a finite number of interior subdomains. This degenerate situation was dealt with by the first time in [GL98]. These results provide us with the first step to obtain the existence of metasolutions of order 1 ≤ k ≤ m by slightly modifying the domain Ω_{k} along the components of its boundary where a(x) vanishes, but this analysis will be done in Section 5, not in Section 3.

In Section 4 we use the results of Section 3 to characterize the limiting behaviour of the regular positive solution as λ ↑ σ_{1}. It will be shown that the regular positive solution is point-wise increasing towards the minimal metasolution of order one of (1.5) supported in Ω_{1} (for λ = σ_{1}).

In Section 5 we use the theory of Section 3 to show that if k ≤ m − 1, then (1.5) exhibits a metasolution of order k supported in Ω_{k} if, and only if, λ < σ_{k+1}, and that in case k = m (1.5) possesses a metasolution of order m supported in Ω_{m} = Ω_{+} for each λ ∈ ℝ, as well as to prove that the point-wise limit of the minimal metasolution of order k ≤ m − 1 as λ ↑ σ_{k+1} provides us with a metasolution of order k + 1 supported in Ω_{k+1} for the value of the parameter λ = σ_{k+1}.

In Section 6 we combine the previous results with the parabolic interior estimates obtained in [Re82] and [Re86] to show that for any 1 ≤ k ≤ m − 1, λ ∈ [σ_{k}, σ_{k+1}), and u_{0} ∈ U_{0}, u_{0} > 0, the restriction of the orbit of u_{[λ,a,Ω]}(·, t; u_{0}), t ≥ 0, to any compact subset K of Ω_{k} is relatively compact in C^{2}(K) with its ω-limit set contained in the closed interval whose ends are the minimal and the maximal regular positive solutions of (1.11), whereas

lim_{t↑∞} u_{[λ,a,Ω]}(·, t; u_{0}) = ∞ uniformly in ∪_{i=1}^{k} ∪_{j=1}^{n_{i}}Ω̄_{0,j}^{i} .

Therefore, if (1.11) possesses a unique regular solution, say θ, then u_{[λ,a,Ω]}(·, t; u_{0}) stabilizes to the metasolution defined by θ as t ↑ ∞.

Most of the available results about the uniqueness of large solutions were obtained in domains where a(x) is bounded away from zero and for rather special classes of nonlinearities (cf. [BM91], [LM93], [LM94], [MV97], and the references there in). One of the main results of [DH99] establishes that if m = 1, n_{1} = 1, Ω̄_{0,1}^{1} ⊂ Ω, and there exist positive constants α and c such that

lim_{d(x)↓0} \frac{a(x)}{d^{α}(x)} = c , d(x) := dist(x, ∂Ω_{0,1}^{1}) ,

then (1.10) possesses a unique regular solution. The idea of the proof consists in combining an extension of Theorem I in [Ke57] with a very well known uniqueness device coming from [BO86]. This seems to be the sole uniqueness result for the case when the coefficient is not bounded away from zero in the totality of Ω. We shall analyze the uniqueness problem in a forthcoming paper.

It should be pointed out that in order to prove most of the results of this paper we need assuming that in any compact subset K of Ω_{+} the following estimate holds

a(x)uf(x, u) − λu ≥ h_{K}(u)

for some function h_{K}(u) satisfying (1.14). This assumption is far from being of technical nature. In fact, if it fails one would need to impose some additional condition on the size of the domain to get the existence of large solutions even in the most simple situations, but this analysis is out of the scope of this work.

The results contained in this paper were communicated in a talk given by the author in Miami during the celebration of the Conference honoring Alan C Lazer (January/9th/1999). The paper was concluded and sent to the journal on August 1999.

2. Regular positive steady states.

In this section we characterize the existence, prove the uniqueness and analyze the point-wise growth of the regular positive solutions of (1.5), as well as their global attractive character with respect to the positive solutions of (1.1).

In the sequel, given a function w ∈ C(Ω̄) we say that w > 0 if w ≥ 0 and w ≠ 0. Given w ∈ U we say that w ≫ 0 if w lies in the interior of U^{+}.

Theorem 2.1. Assume (Ha1-3) and (Hf). Then, the following assertions are true:

(i) The problem (1.5) possesses a regular positive solution if, and only if,

σ_{0} < λ < σ_{1} .  (2.1)

Moreover, it is unique if it exists. In the sequel it will be denoted by θ_{[λ,a,Ω]}. (ii) The map

\begin{matrix}(σ_{0}, σ_{1}) & ⟼ & U \\ λ & → & θ_{[λ,a,Ω]} \end{matrix}

is point-wise increasing and differentiable. Moreover, ∂_{λ}θ_{[λ,a,Ω]} ∈ U^{+}. (iii) Suppose (2.1). Then,

lim_{λ↓σ_{0}} ∥θ_{[λ,a,Ω]}∥_{C(Ω̄)} = 0 ,  (2.2)

and

lim_{λ↑σ_{1}} θ_{[λ,a,Ω]} = ∞ uniformly in ∪_{j=1}^{n_{1}} Ω̄_{0,j}^{1} \ ∂Ω .  (2.3)

Proof. (i) Let u be a regular positive solution of (1.5). Then, u|_{∂Ω} = 0 and

(−Δ + af(·, u))u = λu .

Hence, u is a positive eigenfunction associated with the eigenvalue λ of the operator −Δ + af(·, u_{0}) under homogeneous Dirichlet boundary conditions. Thus, by the uniqueness of the principal eigenvalue (e.g. [Am76]),

λ = σ^{Ω}[−Δ + af(·, u)] ,  (2.4)

and hence, by the monotonicity of the principal eigenvalue with respect to the potential

λ > σ^{Ω}[−Δ] = σ_{0} ,

because a > 0, u ≫ 0 and f(x, u(x)) > 0 for all x ∈ Ω. Moreover, since a = 0 in Ω_{0,1}^{1}, we find from (2.4) that

λ = σ_{1}^{Ω}[−Δ + af(·, u)] < σ^{Ω_{0,1}^{1}} = σ_{1} ,

by the monotonicity of the principal eigenvalue with respect to the domain. Therefore, (2.1) is necessary for the existence of a regular positive solution of (1.5). In order to show that condition (2.1) is sufficient for the existence of a regular positive solution we use the method of sub and supersolutions. Suppose (2.1) and let φ denote the principal eigenfunction associated with σ_{0}. Then, thanks to (Hf), for any ε > 0 sufficiently small the function εφ provides us with a positive subsolution of (1.5). To complete the proof of the existence it remains to show the existence of a supersolution ū of (1.5) such that εφ ≤ ū. For any δ > 0 sufficiently small we consider the δ-neighborhood of Ω_{0} in Ω

Ω_{0}^{δ} := { x ∈ Ω : dist(x, Ω_{0}) < δ } .

Then,

Ω_{0}^{δ} = ∪_{i=1}^{m} ∪_{j=1}^{n_{i}} Ω_{0,j}^{i,δ} ,  (2.5)

where

Ω_{0,j}^{i,δ} := { x ∈ Ω : dist(x, Ω_{0,j}^{i}) < δ } , 1 ≤ i ≤ m , 1 ≤ j ≤ n_{i} .

Thanks to (Ha3), Ω̄_{0,j}^{i} ∩ Ω̄î ; 0,ĵ = ∅ if (i, j) ≠ (î, ĵ), and hence δ > 0 can be chosen sufficiently small so that

Ω̄_{0,j}^{i,δ} ∩ Ω̄_{0,ĵ}^{î,δ} = ∅ if (i, j) ≠ (î, ĵ) .  (2.6)

In the sequel we shall assume that δ > 0 has been chosen in this way.

Since a > 0, and hence Ω_{+} ≠ ∅, for each 1 ≤ i ≤ m and 1 ≤ j ≤ n_{i} the component Ω_{0,j}^{i} is a proper subdomain of Ω. Thus, ∂Ω_{0,j}^{i} ∩ Ω ≠ ∅, and so Ω_{0,j}^{i} is a proper subdomain of Ω_{0,j}^{i,δ/2} and Ω_{0,j}^{i,δ/2} is a proper subdomain of Ω_{0,j}^{i,δ}. Therefore, by the monotonicity of the principal eigenvalue with respect to the domain, we find that

σ^{Ω_{0,j}^{i,δ}} < σ^{Ω_{0,j}^{i,δ/2}} < σ^{Ω_{0,j}^{i}} , 1 ≤ i ≤ m , 1 ≤ j ≤ n_{i} .  (2.7)

On the other hand, thanks to (Ha1), if Γ_{0} and Γ are components of ∂Ω_{0} and ∂Ω, respectively, such that Γ_{0} ∩ Γ ≠ ∅, then Γ_{0} = Γ. Thus, it follows from (2.5) and (2.6) that

∂Ω_{0}^{δ} \ ∂Ω ⊂ Ω_{+} .  (2.8)

Moreover, for each 1 ≤ i ≤ m and 1 ≤ j ≤ n_{i} we have that

lim_{δ↓0} Ω_{0,j}^{i,δ} = Ω_{0,j}^{i} ,

e.g. in the sense of Definition 4.1 of [Lo96]. Thus, it follows from Theorem 4.2 of [Lo96] that

lim_{δ↓0} σ^{Ω_{0,j}^{i,δ}} = σ^{Ω_{0,j}^{i}} , 1 ≤ i ≤ m , 1 ≤ j ≤ n_{i} .  (2.9)

Therefore, thanks to (1.3), (2.1), (2.7) and (2.9), we find that for each δ > 0 sufficiently small

λ < σ^{Ω_{0,j}^{1,δ}} < σ^{Ω_{0,ĵ}^{i,δ}} , 1 ≤ j ≤ n_{1} , 2 ≤ i ≤ m , 1 ≤ ĵ ≤ n_{i} .  (2.10)

Now, for any 1 ≤ i ≤ m and 1 ≤ j ≤ n_{i} let φ_{0,j}^{i,δ} ≫ 0 denote the principal

eigenfunction associated with σ^{Ω_{0,j}^{i,δ}}, and consider the function

Φ(x) := \{ \begin{matrix}φ_{0,j}^{i,δ}(x) if x ∈ Ω_{0,j}^{i,δ/2} for some 1 ≤ i ≤ m , 1 ≤ j ≤ n_{i} , \\ ψ(x) if x ∈ Ω̄ \ Ω_{0}^{δ/2} , \end{matrix}  (2.11)

where ψ(x) is any regular extension of the functions φ_{0,j}^{i,δ} outside Ω_{0}^{δ/2}. Thanks to (2.8), we can assume that ψ is positive and bounded away from zero in Ω̄ \ Ω_{0}^{δ/2}.

We claim that if κ > 1 is sufficiently large, then the function

ū := κΦ

is a supersolution of (1.5) satisfying εφ ≤ ū for ε > 0 small enough. Indeed, in the set

Ω̄ \ Ω_{0}^{δ/2} ⊂ Ω̄_{+}

the function ψ is positive and bounded away from zero, as well as a(x), by (Ha1). Thus, for κ large

−Δψ ≥ λ ψ − aψf(·, κψ) in Ω̄ \ Ω_{0}^{δ/2} ,

since for each x ∈ Ω̄ \ Ω_{0}^{δ/2}

f(x, κψ(x)) ≥ f(x, κ inf_{Ω̄\Ω_{0}^{δ/2}} ψ)

and

lim_{κ↑∞} f(·, κ inf_{Ω̄\Ω_{0}^{δ/2}} ψ) = ∞

uniformly in Ω̄ \ Ω_{0}^{δ/2}. Moreover, thanks to (2.10), in each of the components of Ω_{0}^{δ/2}, Ω_{0,j}^{i,δ/2}, 1 ≤ i ≤ m, 1 ≤ j ≤ n_{i}, we have that for any κ > 0

−Δ(κΦ) = κ σ^{Ω_{0,j}^{i,δ}}φ_{0,j}^{i,δ} > λ κ φ_{0,j}^{i,δ} ≥ λ κ φ_{0,j}^{i,δ} − a κ φ_{0,j}^{i,δ}f(·, κ φ_{0,j}^{i,δ}) .

Therefore, kΦ provides us with a positive supersolution of (1.5) if κ is sufficiently large.

Let Γ be a component of ∂Ω. If Γ is a component of Ω_{+}, then by the construction itself we have that Φ is positive and bounded away from zero on Γ, while if Γ∩∂Ω_{+} = ∅, then Φ = 0 on Γ and \frac{∂Φ}{∂n}(x) < 0 for all x ∈ Γ, where n is the outward unit normal. Therefore, εφ ≤ κΦ provided ε > 0 is sufficiently small. This shows that (2.1) is sufficient for the existence of a regular positive solution of (1.5).

We now show the uniqueness of the positive solution. Suppose (2.1) and let u, v be two positive solutions of (1.5), u ≠ v. Then,

(−Δ + ag − λ)(u − v) = 0 in Ω , u − v = 0 on ∂Ω ,  (2.12)

where

g(x) := \{ \begin{matrix}\frac{u(x)f(x,u(x))−v(x)f(x,v(x))}{u(x)−v(x)} & if u(x) ≠ v(x) , \\ f(x, u(x)) & if u(x) = v(x) , \end{matrix} x ∈ Ω̄ .

By the monotonicity of f on its second argument, it is easily seen that

g > f(·, u) in Ω ,

since u ≠ v. Thus, it follows from (2.4) and the monotonicity of the principal eigenvalue with respect to the potential that

σ^{Ω}[−Δ + a g − λ] ≥ σ^{Ω}[−Δ + af(·, u) − λ] = 0 .

Note that it might happen g = f(·, u) in Ω_{+}, and hence in the previous inequality we should not substitute ≥ by > without some additional work. Assume

σ^{Ω}[−Δ + a g − λ] > 0 .

Then, zero can not be an eigenvalue of −Δ + a g − λ, and hence we find from (2.12) that u = v, which is impossible. Thus,

σ^{Ω}[−Δ + a g − λ] = 0 .

Moreover, u ≠ v. Hence, it follows from (2.12) that there exists κ ∈ ℝ \ {0} such that

u − v = κφ ,

where φ ≫ 0 stands for the principal eigenfunction associated with σ^{Ω}[−Δ + a g − λ] = 0. Therefore, either u(x) < v(x) for all x ∈ Ω, or u(x) > v(x) for all x ∈ Ω. In any of these situations we have that g > f(·, u) in Ω_{+} and therefore

σ^{Ω}[−Δ + a g − λ] > 0 ,

which implies u = v. This contradiction shows the uniqueness and concludes the proof of Part (i).

(ii) Consider the operator F : ℝ × U^{+} → C^{μ}(Ω̄) defined by

F(λ, u) := −Δu − λu + auF (·, u) , (λ, u) ∈ ℝ × U^{+} ,

where F stands for the substitution operator induced by f, and pick up λ ∈ (σ_{0}, σ_{1}). Then, F is an operator of class C^{1} in the interior of U^{+} such that

F(λ, θ_{[λ,a,Ω]}) = 0 ,

and

D_{u}F(λ, θ_{[λ,a,Ω]}) = −Δ − λ + af(·, θ_{[λ,a,Ω]}) + aθ_{[λ,a,Ω]}∂_{u}f(·, θ_{[λ,a,Ω]}) .

By (Hf), (2.4) and the monotonicity of the principal eigenvalue with respect to the potential we find that

σ^{Ω}[D_{u}F(λ, θ_{[λ,a,Ω]})] > σ^{Ω}[−Δ − λ + af(·, θ_{[λ,a,Ω]})] = 0 .  (2.13)

Therefore, D_{u}F(λ, θ_{[λ,a,Ω]}) is an isomorphism and it follows from the implicit function theorem that there exist ε > 0 and a map of class C^{1}, u : (λ − ε, λ + ε) → U^{+}, such that u(λ) = θ_{[λ,a,Ω]} and for each s ∈ (λ − ε, λ + ε)

F(s, u(s)) = 0 .

Moreover, those are the unique zeroes of F in a neighborhood of (λ, θ_{[λ,a,Ω]}) in ℝ × U^{+}, and u(s) ≫ 0, since u(λ) ≫ 0. Thus, thanks to the uniqueness of the positive solution,

u(s) = θ_{[s,a,Ω]} , s ≃ λ .

Furthermore, by implicit differentiation we find that

D_{u}F(λ, θ_{[λ,a,Ω]})∂_{λ}θ_{[λ,a,Ω]} = θ_{[λ,a,Ω]} ≫ 0 .  (2.14)

Thanks to (2.13) the differential operator on the left hand side of (2.14) satisfies the strong maximum principle. Therefore, (2.14) implies

∂_{λ}θ_{[λ,a,Ω]} ≫ 0 .

This completes the proof of Part (ii).

(iii) Relation (2.2) follows from the uniqueness of the positive solution taking into account that thanks to the main theorem of [CR71] λ = σ_{0} is a bifurcation value to positive solutions of (1.5) from u = 0. We now show (2.3). Set

Ω_{0}^{1} := ∪_{j=1}^{n_{1}}Ω_{0,j}^{1} .

Since a = 0 in Ω_{0}^{1}, (2.14) gives

(−Δ − λ)∂_{λ}θ_{[λ,a,Ω]} = θ_{[λ,a,Ω]} in Ω_{0}^{1} , λ ∈ (σ_{0}, σ_{1}) .

Now, pick λ̂ ∈ (σ_{0}, σ_{1}) and consider c > 0 such that for each 1 ≤ j ≤ n_{1},

θ_{[λ̂,a,Ω]} > cφ_{0,j}^{1} in Ω_{0,j}^{1} .

Recall that φ_{0,j}^{1} is the principal eigenfunction associated with σ^{Ω_{0,j}^{1}}. Thanks to Part (ii), for each λ ∈ (λ̂, σ_{1}) we have that

θ_{[λ,a,Ω]} > θ_{[λ̂,a,Ω]} > cφ_{0,j}^{1} in Ω_{0,j}^{1} , 1 ≤ j ≤ n_{1} .

Moreover, for each λ ∈ (λ̂, σ_{1}) and 1 ≤ j ≤ n_{1} the operator −Δ − λ satisfies the strong maximum principle in Ω_{0,j}^{1}. Hence,

∂_{λ}θ_{[λ,a,Ω]} > c (−Δ − λ)^{−1}φ_{0,j}^{1} = \frac{c}{σ_{1} − λ}φ_{0,j}^{1} in Ω_{0,j}^{1} .

Note that, thanks to (Ha3), σ_{1} = σ^{Ω_{0,j}^{1}} for each 1 ≤ j ≤ n_{1}. On the other hand, for each 1 ≤ j ≤ n_{1}, the function φ_{0,j}^{1} is bounded away from zero on any compact subset of Ω_{0,j}^{1}. Thus,

lim_{λ↑σ_{1}} ∂_{λ}θ_{[λ,a,Ω]} = ∞ uniformly in compact subsets of Ω_{0}^{1} ,

and therefore

lim_{λ↑σ_{1}} θ_{[λ,a,Ω]} = ∞ uniformly in compact subsets of Ω_{0}^{1} .

It remains to show that

lim_{λ↑σ_{1}} θ_{[λ,a,Ω]}(x) = ∞ for all x ∈ ∂Ω_{0}^{1} \ ∂Ω .  (2.15)

For this, consider δ > 0 sufficiently small, pick λ satisfying

σ^{Ω_{0,j}^{1,δ}} < σ^{Ω_{0,j}^{1,δ/2}} < λ < σ^{Ω_{0,j}^{1}} = σ_{1} , 1 ≤ j ≤ n_{1} ,

and introduce the function u_{δ} ∈ C(Ω̄) defined by

u_{δ}(x) := \{ \begin{matrix}C φ_{0,j}^{1,δ}(x) & if x ∈ Ω̄_{0,j}^{1,δ/2} for some 1 ≤ j ≤ n_{1} , \\ 0 & if x ∈ Ω̄ \ ∪_{j=1}^{n_{1}}Ω_{0,j}^{1,δ/2} , \end{matrix}  (2.16)

where C > 0 is a positive constant. Then, the argument of the proof of Theorem 4.3 in [LS98] can be easily adapted to show that under condition (Ha2) there exists C = C(δ) > 0 such that u_{δ} is a subsolution of (1.5) satisfying

lim_{δ↓0} u_{δ}(x) = ∞

for each x ∈ ∂Ω_{0}^{1} \ ∂Ω. By the uniqueness of the positive solution, necessarily u_{δ} ≤ θ_{[λ,a,Ω]} and hence, (2.15) is satisfied. The uniform divergence in ∂Ω_{0}^{1} follows from the point-wise monotonicity in λ as an immediate consequence from Dini’s theorem. This concludes the proof of the theorem. □

We shall say that a non-negative positive steady-state u of (1.1), i.e. a non-negative solution of (1.5), is globally asymptotically stable if

lim_{t↑∞} ∥u_{[λ,a,Ω]}(·, t; u_{0}) − u∥_{C(Ω̄)} = 0

for each u_{0} ∈ U_{0}, u_{0} > 0. The main result on the longtime behaviour of the positive solutions of (1.1) reads as follows.

Theorem 2.2. Under the assumptions of Theorem 2.1, the following assertions are true:

(i) If λ ≤ σ_{0}, then u = 0 is globally asymptotically stable.

(ii) If σ_{0} < λ < σ_{1}, then θ_{[λ,a,Ω]} is globally asymptotically stable.

(iii) Set

Ω_{0}^{i} := ∪_{j=1}^{n_{i}}Ω_{0,j}^{i} , 1 ≤ i ≤ m ,  (2.17)

and assume σ_{i} ≤ λ for some 1 ≤ i ≤ m. Then, for any u_{0} ∈ U_{0} with u_{0} > 0 we have

lim_{t↑∞} u_{[λ,a,Ω]}(·, t; u_{0}) = ∞ uniformly in ∪_{k=1}^{i} Ω̄_{0}^{k} \ ∂Ω .  (2.18)

Proof. Recall that in the proof of Theorem 2.1(i) we have shown that (1.5) possesses arbitrarily large supersolutions in the interior of U^{+} for any λ < σ_{1}.

(i) Let u_{0} ∈ U_{0} such that u_{0} > 0 and consider the solution u(x, t; u_{0}) of (1.1). By the parabolic maximum principle for any t > 0 the function u(·, t; u_{0}) lies in the interior of U^{+}. Fix t_{1} > 0 and consider a supersolution ū ≫ 0 of (1.5) such that

u(·, t_{1}; u_{0}) ≪ ū .

Then, for all t ≥ t_{1} we have that

u(·, t; u_{0}) ≪ u(·, t − t_{1}; ū) .  (2.19)

Thanks to the results of [Sa73], the function t → u(·, t−t_{1}; ū) is decreasing and converges to a non-negative solution of (1.5). Due to Theorem 2.1(i) it must converge to zero. Therefore, (2.19) completes the proof of Part (i).

(ii) Let u_{0} ∈ U_{0} be such that u_{0} > 0 and pick t_{1} > 0. Since σ_{0} < λ < σ_{1}, there exist a subsolution \underline{u} and a supersolution ū of (1.5), both in the interior of U^{+}, such that

\underline{u} ≪ u(·, t_{1}; u_{0}) ≪ ū .

Then, for all t ≥ t_{1} we have that

u(·, t − t_{1}; \underline{u}) ≪ u(·, t; u_{0}) ≪ u(·, t − t_{1}; ū) .  (2.20)

Thanks to the results of [Sa73], the function t → u(·, t − t_{1}; ū) is decreasing and converges to a non-negative solution of (1.5), whereas t → u(·, t−t_{1}; \underline{u}) is increasing and converges to a positive solution of (1.5). By the uniqueness of the positive solution, both functions must converge to θ_{[λ,a,Ω]}. Combining these features with (2.20) concludes the proof of this part.

(iii) Assume that λ ≥ σ_{i} for some 1 ≤ i ≤ m and consider u_{0} ∈ U_{0} with u_{0} > 0. Let u = u_{[λ,a,Ω]}(x, t; u_{0}) be the unique global regular solution of (1.1). For all ε > 0 we have λ > σ_{1} − ε, and hence

∂_{t}u − Δu = λu − auf(x, u) > (σ_{1} − ε)u − auf(x, u) .

Thus, for each t > 0

u_{[λ,a,Ω]}(·, t; u_{0}) ≫ u_{[σ_{1}−ε,a,Ω]}(·, t; u_{0}) .  (2.21)

By Part (ii), we already know that

lim_{t↑∞} ∥u_{[σ_{1}−ε,a,Ω]}(·, t; u_{0}) − θ_{[λ,a,Ω]}∥_{C(Ω̄)} = 0 .

Hence, it follows from (2.21) that for each ε > 0

lim inf_{t↑∞} u_{[λ,a,Ω]}(·, t; u_{0}) ≥ θ_{[σ_{1}−ε,a,Ω]} .  (2.22)

Thanks to Theorem 2.1(iii),

lim_{ε↓0} θ_{[σ_{1}−ε,a,Ω]} = ∞ uniformly in Ω̄_{0}^{1} \ ∂Ω .

Therefore, we find from (2.22) that

lim inf_{t↑∞} u_{[λ,a,Ω]}(·, t; u_{0}) = ∞ uniformly in Ω̄_{0}^{1} \ ∂Ω .

This completes the proof if i = 1.

Assume i ≥ 2 and

lim inf_{t↑∞} u_{[λ,a,Ω]}(·, t; u_{0}) = ∞ uniformly in ∪_{k=1}^{i−1} Ω̄_{0}^{k} \ ∂Ω .

Let â ∈ C^{μ}(Ω̄) any weight function such that â > a and â(x) > 0 if, and only if, x ∈ Ω_{+} ∪ ∪_{k=1}^{i−1}Ω̄_{0}^{k}. Then, the lower order refuges of â are Ω_{0,j}^{i}, 1 ≤ j ≤ n_{i}. Recall that we are assuming

λ ≥ σ_{i} = σ^{Ω_{0,1}^{i}} .

Since â > a, we have

u_{[λ,a,Ω]}(·, t; u_{0}) ≫ u_{[λ,â,Ω]}(·, t; u_{0}) .

Moreover, by the corresponding result for the case i = 1,

lim inf_{t↑∞} u_{[λ,â,Ω]}(·, t; u_{0}) = ∞ uniformly in Ω̄_{0}^{i} \ ∂Ω .

This completes the proof of the theorem. □

3. A priori interior estimates and the existence of large regular solutions.

Beside their own interest, the following results are crucial to show the stabilization in Ω \ Ω̄_{0}^{1} of the regular positive solutions of (1.5) as λ ↑ σ_{1}. They also show the existence of large regular solutions of

−Δu = λu − auf(·, u)

in Ω. By a large regular solution we mean a solution of class C^{2+μ} which grows arbitrarily when the spatial variable approaches to some of the components of ∂Ω. Those solutions will provide us with the limiting behaviour of the population as times grows to infinity for any λ ≥ σ_{1}.

Theorem 3.1. Suppose (Ha1-3), (Hf) and

∂Ω ∩ ∂Ω_{+} ≠ ∅ .

Let Γ_{+}^{j}, 1 ≤ j ≤ q, be q arbitrary components of ∂Ω ∩ ∂Ω_{+}, and α_{j} > 0, 1 ≤ j ≤ q, q arbitrary constants, and consider the nonlinear boundary value problem

−Δu = λu − a(x)uf(x, u) in Ω ,
u|_{Γ_{+}^{j}} = α_{j} > 0 , 1 ≤ j ≤ q ,
u = 0 on ∂Ω \ ∪_{j=1}^{q}Γ_{+}^{j} .

(3.1)

The following assertions are true:

(i) Problem (3.1) possesses a regular positive solution if, and only if, λ < σ_{1}. Moreover, it is unique if it exists. In the sequel we shall denote it by Θ_{[λ,a,Ω,Γ_{+},α]} where

Γ_{+} := (Γ_{+}^{1}, …, Γ_{+}^{q}) , α := (α_{1}, …, α_{q}) .

(ii) The map (−∞, σ_{1}) → C(Ω̄), λ → Θ_{[λ,a,Ω,Γ_{+},α]} is point-wise increasing, as well as the map (0, ∞)^{q} → C(Ω̄), α = (α_{1}, …, α_{q}) → Θ_{[λ,a,Ω,Γ_{+},α]}. Moreover,

lim_{λ↑σ_{1}} Θ_{[λ,a,Ω,Γ_{+},α]} = ∞ uniformly in ∪_{j=1}^{n_{1}} Ω̄_{0,j}^{1} \ ∂Ω .  (3.2)

Proof. (i) Suppose (3.1) possesses a regular positive solution, say u. Then,

(−Δ + af(·, u) − λ)u = 0 in Ω ,

and u|_{∂Ω} > 0. Thus, u provides us with a positive strict supersolution of −Δ + af(·, u) − λ in Ω under homogeneous Dirichlet boundary conditions. Hence, it follows from the characterization of the strong maximum principle found in [LM94] (cf. [Lo96] and [AL98]) that

σ^{Ω}[−Δ + af(·, u) − λ] > 0 .  (3.3)

Therefore,

λ < σ^{Ω}[−Δ + af(·, u)] < σ^{Ω_{0,1}^{1}} = σ_{1} ,

since a = 0 in Ω_{0,1}^{1}.

Suppose λ < σ_{1}. To show that (3.1) possesses a regular positive solution we use the method of sub and supersolutions. The function \underline{u} := 0 provides us with a subsolution of (3.1). Moreover, we can use the same supersolutions constructed in the proof of Theorem 2.1. Indeed, to show that κΦ provides us with a positive supersolution of (3.1), where Φ is the function defined in (2.11), it remains to check that for any κ sufficiently large

κΦ|_{Γ_{+}^{j}} > α_{j} , 1 ≤ j ≤ q .  (3.4)

By construction,

∪_{j=1}^{q}Γ_{+}^{j} ⊂ Ω̄ \ Ω_{0}^{δ/2} ,

and hence

κΦ|_{Γ_{+}^{j}} = κψ|_{Γ_{+}^{j}} , 1 ≤ j ≤ q .

Moreover, ψ is positive and bounded away from zero in Ω̄ \ Ω_{0}^{δ/2}. Therefore, (3.4) holds provided κ is large enough. This concludes the proof of the existence.

The uniqueness of the regular positive solution follows from (3.3) with the same argument used to show the uniqueness in the proof of Theorem 2.1. This completes the proof of Part (i).

(ii) Let λ_{1}, λ_{2} ∈ (σ_{0}, σ_{1}) such that λ_{1} < λ_{2}. Then, setting

Θ_{i} := Θ_{[λ_{i},a,Ω,Γ_{+},α]} , i = 1 , 2 ,

we find from their definition that

(−Δ + a g − λ_{1})(Θ_{2} − Θ_{1}) > 0 in Ω , (Θ_{2} − Θ_{1})|_{∂Ω} = 0 .  (3.5)

where

g(x) := \{ \begin{matrix}\frac{Θ_{2}(x)f(x, Θ_{2}(x)) − Θ_{1}(x)f(x, Θ_{1}(x))}{Θ_{2}(x) − Θ_{1}(x)} & if Θ_{2}(x) ≠ Θ_{1}(x) , x ∈ Ω̄ , \\ f(x, Θ_{2}(x)) & if Θ_{2}(x) = Θ_{1}(x) , x ∈ Ω̄ . \end{matrix}

By the monotonicity of f on its second argument, it is easily seen that

g > f(·, Θ_{1}) in Ω ,

since Θ_{2} ≠ Θ_{1}. Thus, thanks to (3.3), it follows from the monotonicity of the principal eigenvalue with respect to the potential that

σ^{Ω}[−Δ + a g − λ_{1}] ≥ σ^{Ω}[−Δ + af(·, Θ_{1}) − λ_{1}] > 0 .

Henceforth, the operator −Δ + a g − λ_{1} satisfies the strong maximum principle in Ω under homogeneous Dirichlet boundary conditions, and therefore we find from (3.5) that

Θ_{2} − Θ_{1} ≫ 0 .

This completes the proof of the monotonicity in λ. The same argument can be easily adapted to get the monotonicity in α.

Relation (3.2) follows easily from Theorem 2.1 taking into account that for any λ ∈ (σ_{0}, σ_{1})

Θ_{[λ,a,Ω,Γ_{+},α]} ≫ θ_{[λ,a,Ω]} .

The proof is completed. □

Remark 3.2: (a) Theorem 3.1 also holds in the absence of refuges, i.e. in case Ω = Ω_{+}. In this special situation its proof is simpler, since large constants provide us with supersolutions for any λ ∈ ℝ. Therefore, in case Ω = Ω_{+} the positive solution exists for each λ ∈ ℝ. This is the case dealt with in most of the references and in particular in [MV97] and [DH99] for the special choice (1.7).

(b) By the uniqueness of the positive solution, any couple (\underline{u}, \overline{u}) formed by a nonnegative subsolution \underline{u} and a nonnegative supersolution \overline{u} of (3.1) must satisfy

\underline{u} ≤ Θ_{[λ,a,Ω,Γ_{+},α]} ≤ \overline{u} .

This estimate can be easily obtained with the same comparison argument used to show the uniqueness and the monotonicities, and so we will omit the details of its proof.

In the sequel we shall assume the following:

(Hfb) There exists a continuous function f_{b} : [0, ∞) → ℝ of class C^{1+μ}((0, ∞)) such that f_{b}(0) = 0, f_{b}(u) > 0 and f′_{b}(u) > 0 for all u > 0,

lim_{u↑∞} f_{b}(u) = ∞ ,

and

f(·, u) ≥ f_{b}(u) , u ≥ 0 .

The following result is a substantial generalization of the interior estimates found in [Ke57] and [Os57]. It provides us with uniform interior estimates in Ω_{+} for the solutions of (3.1) under assumption (Hfb), and we shall we use it to show the stabilization in Ω_{+} of Θ_{[λ,a,Ω,Γ_{+},α]} as either α_{j} ↑ ∞, 1 ≤ j ≤ q, or λ ↑ σ_{1}, and to obtain very general existence results of large regular solutions.

Theorem 3.3. Suppose (Ha1-3), (Hf) and (Hfb). Let D ⊂ Ω_{+} be open and K ⊂ Ω_{+} compact such that D + B_{δ} ⊂ K for some δ > 0, where B_{δ} stands for the ball of radius δ centered at the origin. Set

A := inf_{x∈K} a(x) > 0 ,

fix β > 0 and let u_{K} denote the unique positive zero of the function

h(u) := Auf_{b}(u) − βu .  (3.6)

Thanks to (Hfb), u_{K} is well defined. Assume in addition that for each u_{∗} > u_{K}

I(u_{∗}) := ∫_{u_{∗}}^{∞} [∫_{u_{∗}}^{u} h(z) dz]^{−1/2} du < ∞ .  (3.7)

Then, there exists a universal constant M = M(β) such that for any λ ≤ β and any regular positive solution u_{λ} of

−Δu = λu − auf(·, u) in Ω_{+} ,  (3.8)

the following uniform estimate is satisfied

∥u_{λ}∥_{C(D̄)} ≤ M .  (3.9)

Remarks 3.4: (a) In (3.9) no condition on the growth of u_{λ} on ∂Ω_{+} is imposed. Therefore, (3.9) provides us with a universal interior estimate for the positive solutions of (3.8) which is independent of the behaviour of the solutions on the boundary.

(b) We should point out that

∫_{u_{K}}^{∞} [∫_{u_{K}}^{u} h(z) dz]^{−1/2} du = ∞ .  (3.10)

Indeed, the auxiliary function

g(u) := ∫_{u_{K}}^{u} h(z) dz , u > u_{K} ,

satisfies

g(u_{K}) = 0 , g′(u_{K}) = h(u_{K}) = 0 ,

and

g′′(u_{K}) = h′(u_{K}) = Af_{b}(u_{K}) + Au_{K}f′_{b}(u_{K}) − β = Au_{K}f′_{b}(u_{K}) > 0 .

Therefore, the integral diverges.

(c) If there exist η > 0 and p > 0 such that

f_{b}(u) ≥ η u^{p} , u ≥ 0 ,

then condition (3.7) is satisfied for any A > 0 and β > 0, and therefore the estimate (3.9) is valid in any open set D satisfying D̄ ⊂ Ω_{+}. Indeed, for any u_{∗} > u_{K} we have that

\begin{matrix}I(u_{∗}) = & \sqrt{2} ∫_{u_{∗}}^{∞} [2A ∫_{u_{∗}}^{u} zf_{b}(z) dz − β(u^{2} − u_{∗}^{2})]^{−1/2} du \\ = & \sqrt{2} ∫_{1}^{∞} [2A ∫_{1}^{θ} tf_{b}(u_{∗}t) dt − β(θ^{2} − 1)]^{−1/2} dθ . \end{matrix}

In particular, I(u_{∗}) is decreasing and

I(u_{∗}) ≤ \sqrt{2} ∫_{1}^{∞} [\frac{2Aη}{p + 2}u_{∗}^{p}(θ^{p+2} − 1) − β(θ^{2} − 1)]^{−1/2} dθ < ∞ .

(d) Condition (3.7) entails the existence of a unique regular positive solution of

−u′′ = βu − Auf_{b}(u) in (−R, R) , u(−R) = u(R) = ∞ ,  (3.11)

for each R > 0. Indeed, multiplying the differential equation of (3.11) by u′ and setting

v = u′ , u(0) = u_{∗} ,

it is easily seen that the solutions of (3.11) are given by the u_{∗}’s such that for each x ∈ (0, R)

v^{2}(x) = 2 ∫_{u_{∗}}^{u(x)} h(z) dz

and lim_{x↑R} u(x) = ∞. Necessarily u_{∗} > u_{K}, because otherwise u ≤ u_{K}, since u_{K} is an equilibrium of the equation. Moreover, this occurs if and only if

R = \sqrt{2} ∫_{u_{∗}}^{∞} [∫_{u_{∗}}^{u} h(z) dz]^{−1/2} du .

Therefore, u_{∗} corresponds to a solution of (3.11) if and only if u_{∗} > u_{K} and

R = \sqrt{2}I(u_{∗}) .

We already know that lim_{u_{∗}↓u_{K}} I(u_{∗}) = ∞ and that I(u_{∗}) is decreasing. Moreover, since h is nondecreasing, it is easily seen from (3.7) that

lim_{u_{∗}↑∞} I(u_{∗}) = 0 .

Therefore, there exists a unique u_{∗} ∈ (u_{K}, ∞) such that R = \sqrt{2}I(u_{∗}). Actually, this implies the existence of a radially symmetric regular large solution for the problem in a ball (cf. Lemma 6.2 of [GGLS98]). The proof of Theorem 3.3 can be accomplished very easily from these facts, but in this paper we prefer using the techniques introduced in [Ke57], since they are pioneer in the field. If in addition h′′(u) ≥ 0 for u large and

lim_{u↑∞} \frac{h′(u)}{\sqrt{∫_{u_{K}}^{u} h(z) dz}} > 0 ,

then the regular large solution in the ball is unique, [LM94].

(e) Even in the simplest case when there exists p > 0 such that f(x, u) = f_{b}(u) = u^{p} the condition (2) of [Ke57] fails, because of (3.10). This is because the nonlinearity h(u) of [Ke57] is increasing and bounded away from zero. Therefore, the interior estimates obtained in [Ke57] do not guarantee straight away the existence of interior estimates for the positive solutions of (3.1). Among other things this implies that the proof of Lemma 1.3 of [MV97] contains a gap, since the interior estimates of [Ke57] can not be used. Our Theorem 3.3 completes the proof of Lemma 1.3 of [MV97].

Although the proof of Theorem 3.3 is an easy consequence from Theorem III of [Ke57], as pointed out to us by the referee, we are going to give a complete selfcontained proof of it by means of the technical tools introduced in the proof of Theorem I of [Ke57]. Note that in [Ke57] Theorem III was obtained as a consequence from Theorem I.

Proof of Theorem 3.3. It suffices to show that for each x_{0} ∈ D̄ there exists η > 0 and a constant M > 0 such that for any λ ≤ β and any regular positive solution u_{λ} of (3.8)

∥u_{λ}∥_{C(B_{η}(x_{0}))} ≤ M ,

where B_{η}(x_{0}) stands for the ball of radius η centered at x_{0}. Let x_{0} ∈ D̄ and consider R > 0 such that B̄_{R}(x_{0}) ⊂ K. Then, for each λ ≤ β the following differential inequality holds

Δu_{λ} ≥ h(u_{λ}) in B_{R}(x_{0}) ,  (3.12)

where h(u) is the function defined by (3.6). Set

α_{λ} := max{u_{K} + 1 , sup_{x∈∂B_{R}(x_{0})} u_{λ}} ,  (3.13)

and let Θ_{λ} denote the unique positive solution of

Δu = h(u) in B_{R}(x_{0}) , u|_{∂B_{R}(x_{0})} = α_{λ} ,  (3.14)

whose existence is guaranteed by Theorem 3.1 (cf. Remark 3.2(a)). Thanks to (3.12) and (3.13), for each λ ≤ β any solution u_{λ} of (3.8) is a positive subsolution of (3.14) in B_{R}(x_{0}), and hence it follows from Remark 3.2(b) that

u_{λ} ≤ Θ_{λ} in B_{R}(x_{0}) .

Thus, to complete the proof it suffices to show that there exist η ∈ (0, R) and a constant M > 0 such that for any α_{λ} > 0 sufficiently large

∥Θ_{λ}∥_{C(B_{η}(x_{0}))} ≤ M .  (3.15)

By definition, h(u_{K}) = 0 and h(u) > 0 for u > u_{K}. Thus, for any α_{λ} > u_{K} the constant u_{K} is a strict positive subsolution of (3.14), and hence it follows from Remark 3.2(b) that

u_{K} ≤ Θ_{λ} .

In fact, the strong maximum principle implies u_{K} ≪ Θ_{λ}. On the other hand, by the uniqueness of the positive solution of (3.14), Θ_{λ} must be radially symmetric, since the problem is invariant by rotations. Thus,

Θ_{λ}(x) = Ψ_{λ}(|x − x_{0}|) , x ∈ B_{R}(x_{0}) ,

where Ψ_{λ}(r) is the unique positive solution of

ψ′′(r) + \frac{N − 1}{r}ψ′(r) = h(ψ(r)) , 0 < r < R , ψ′(0) = 0 , ψ(R) = α_{λ} . (3.16)

Therefore, to prove (3.15) it suffices to show that there exist η > 0 and M > 0 such that for any α_{λ} > u_{K}

∥Ψ_{λ}∥_{C([0,η])} ≤ M .  (3.17)

We already know that Ψ_{λ}(r) > u_{K} for each r ∈ [0, R] and α_{λ} > u_{K}. Hence,

h(Ψ_{λ}(r)) > 0 .

Moreover, for each u > u_{K} we have that

h′(u) = Af_{b}(u) + Auf′_{b}(u) − β > Af_{b}(u_{K}) − β + Auf′_{b}(u) = Auf′_{b}(u) > 0 ,

and hence h is increasing. We now follow the proof of Theorem I in page 506 of [Ke57]. The functions Ψ_{λ} satisfy

(R^{N−1}Ψ′_{λ}(r))′ = r^{N−1}h(Ψ_{λ}(r)) ,  (3.18)

and hence integrating (3.18) from 0 to r yields

Ψ′_{λ}(r) = r^{1−N} ∫_{0}^{r} s^{N−1}h(Ψ_{λ}(s)) ds > 0 .  (3.19)

This shows that r → Ψ_{λ}(r) is increasing, as well as r → h(Ψ_{λ}(r)). Thus, we find from (3.19) that

Ψ′_{λ}(r) ≤ r^{1−N}h(Ψ_{λ}(r)) ∫_{0}^{r} s^{N−1} ds = \frac{r}{N} h(Ψ_{λ}(r)) .  (3.20)

Now, substituting (3.20) into (3.16) gives

Ψ′′_{λ} ≥ \frac{h(Ψ_{λ})}{N} .

Moreover, since Ψ′_{λ} ≥ 0, (3.16) gives Ψ′′_{λ} ≤ h(Ψ_{λ}). Hence,

h(Ψ_{λ}) ≥ Ψ′′_{λ} ≥ \frac{h(Ψ_{λ})}{N} .  (3.21)

We now multiply (3.21) by Ψ′_{λ} and integrate from 0 to r to obtain

2 ∫_{Ψ_{λ}(0)}^{Ψ_{λ}(r)} h(z) dz ≥ [Ψ′_{λ}(r)]^{2} ≥ \frac{2}{N} ∫_{Ψ_{λ}(0)}^{Ψ_{λ}(r)} h(z) dz .  (3.22)

Now, taking the square root of the reciprocal of (3.22) and integrating again gives

\frac{1}{\sqrt{2}}∫_{Ψ_{λ}(0)}^{Ψ_{λ}(r)}[∫_{Ψ_{λ}(0)}^{u} h(z) dz]^{−1/2}du ≤ r ≤ \sqrt{\frac{N}{2}} ∫_{Ψ_{λ}(0)}^{Ψ_{λ}(r)}[∫_{Ψ_{λ}(0)}^{u} h(z) dz]^{−1/2}du .

(3.23)

In particular,

\frac{1}{\sqrt{2}}∫_{Ψ_{λ}(0)}^{α_{λ}}[∫_{Ψ_{λ}(0)}^{u} h(z) dz]^{−1/2}du ≤ R .

Hence, thanks to (3.10), Ψ_{λ}(0) must be uniformly bounded away from u_{K}. Moreover,

R ≤ \sqrt{\frac{N}{2}} ∫_{Ψ_{λ}(0)}^{∞}[∫_{Ψ_{λ}(0)}^{u} h(z) dz]^{−1/2}du .

So, since h in increasing we find from (3.7) that Ψ_{λ}(0) must be uniformly bounded above. Therefore, using Remark 3.2(b) gives

Ψ_{λ}(0) < lim sup_{α_{λ}↑∞} Ψ_{λ}(0) = Ψ_{0} ∈ (u_{K}, ∞) ,

for any α_{λ} > u_{K}. Let Ψ_{∞} denote the unique positive solution of the Cauchy problem

ψ′′(r) + \frac{N − 1}{r}ψ′(r) = h(ψ(r)) 0 < r < R , ψ′(0) = 0 , ψ(0) = Ψ_{0} .

By continuous dependence, Ψ_{∞} is defined in [0, R) and

lim_{r↑R} Ψ_{∞}(r) = ∞ .

Moreover, thanks to Remark 3.2(b), Ψ_{λ} ≤ Ψ_{∞}. This completes the proof. □

Note that Ψ_{∞} is the minimal large solution of Δu = h(u) in the ball B_{R}. The heart of the proof of Theorem 3.3 consists in showing the existence of such large solution. A different proof of this fact was given in [GGLS98] for the special case when f(x, u) = f_{b}(u) = u^{p}.

Combining Theorem 3.1 with Theorem 3.3 it follows the existence of large positive solutions for a wide class of semilinear boundary value problems, even in the presence of refuges. In fact the following result provides us with a substantial extension of most of the results available in the literature.

Theorem 3.5. Suppose (Ha1-3), (Hf), (Hfb) and λ < σ_{1}. Assume in addition that (3.7) is satisfied for any compact subset K ⊂ Ω_{+} and β = σ_{1} and that

∂Ω ∩ ∂Ω_{+} ≠ ∅ .

Let Γ_{+}^{j}, 1 ≤ j ≤ q, be q arbitrary components of ∂Ω∩∂Ω_{+} and consider the singular boundary value problem

−Δu = λu − a(x)uf(x, u) in Ω ,
u = ∞ on γ_{∞} := ∪_{j=1}^{q}Γ_{+}^{j} ,
u = 0 on γ_{0} := ∂Ω \ γ_{∞} .

(3.24)

Then, the following assertions are true:

(i) The problem (3.24) possesses a regular positive solution u. By regular solution we mean u ∈ C^{2+μ}(Ω) and

lim_{dist(x,γ_{∞})↓0} u(x) = ∞ , lim_{dist(x,γ_{0})↓0} u(x) = 0 .

(ii) The function

Θ_{[λ,a,Ω,Γ_{+},∞]}(x) := lim_{α↑∞} Θ_{[λ,a,Ω,Γ_{+},α]}(x) , x ∈ Ω ,  (3.25)

where Θ_{[λ,a,Ω,Γ_{+},α]} is the unique regular positive solution of (3.1), is the minimal regular positive solution of (3.24). Moreover,

lim_{α↑∞} ∥Θ_{[λ,a,Ω,Γ_{+},α]} − Θ_{[λ,a,Ω,Γ_{+},∞]}∥_{C^{2+μ}(Ω)} = 0 .  (3.26)

By α ↑ ∞ we mean α_{j} ↑ ∞ for each 1 ≤ j ≤ q.

(iii) For each x ∈ ∪_{j=1}^{n_{1}}Ω̄_{0,j}^{1} \ ∂Ω

lim_{λ↑σ_{1}} Θ_{[λ,a,Ω,Γ_{+},∞]}(x) = ∞ ,  (3.27)

whereas the function

Θ_{[σ_{1},a,Ω,Γ_{+},∞]}(x) := lim_{λ↑σ_{1}} Θ_{[λ,a,Ω,Γ_{+},∞]}(x) , x ∈ Ω \ ∪_{j=1}^{n_{1}}Ω̄_{0,j}^{1} ,  (3.28)

is well defined and it provides us with the minimal regular positive solution of

−Δu = λu − a(x)uf(x, u) in Ω \ ∪_{j=1}^{n_{1}}Ω̄_{0,j}^{1} ,
u = ∞ on γ_{∞}^{1} := γ_{∞} ∪ (∪_{j=1}^{n_{1}}∂Ω_{0,j}^{1} \ ∂Ω) ,
u = 0 on ∂(Ω \ ∪_{j=1}^{n_{1}}Ω̄_{0,j}^{1}) \ γ_{∞}^{1} .

(3.29)

Proof. Choose δ > 0 sufficiently small so that

K_{j,δ} := {x ∈ Ω : dist(x, Γ_{+}^{j}) ≤ δ} ⊂ Ω_{+} , 1 ≤ j ≤ q ,

and K_{i,δ} ∩ K_{j,δ} = ∅ if i ≠ j, and consider the open set

D := Ω \ ∪_{j=1}^{q}K_{j,δ} .

By construction

D_{+} := {x ∈ D : a(x) > 0} = Ω_{+} \ ∪_{j=1}^{q}K_{j,δ}

and

D̄_{0} := {x ∈ D : a(x) = 0} = Ω̄_{0} .

Moreover,

∂D = γ_{0} ∪ γ_{∞}^{D} ,

where

γ_{∞}^{D} := ∪_{j=1}^{q}{x ∈ Ω_{+} : dist(x, Γ_{+}^{j}) = δ} .

By construction, γ_{∞}^{D} is a compact subset of Ω_{+}, and hence it follows from Theorem 3.3 that there exists a universal constant M > 0 such that

∥Θ_{[λ,a,Ω,Γ_{+},α]}∥_{C(γ_{∞}^{D})} ≤ M

for all α = (α_{1}, …, α_{q}) ∈ (0, ∞)^{q}. Thus, any solution of (3.1) provides us with a subsolution of

\begin{matrix}−Δu = λu − a(x)uf(x, u) in D , \\ u = M on γ_{∞}^{D} , \\ u = 0 on γ_{0} . \end{matrix}  (3.30)

Due to Remark 3.2(b) we have that for each α ∈ (0, ∞)^{q}

Θ_{[λ,a,Ω,Γ_{+},α]} ≤ Θ_{[λ,a,D,γ_{∞}^{D},M]} in D ,  (3.31)

where Θ_{[λ,a,D,γ_{∞}^{D},M]} is the unique positive solution of (3.30), whose existence is guaranteed by Theorem 3.1.

Since δ can be taken arbitrarily small,

lim_{δ↓0} D = Ω

in the sense of [Lo96], and the mapping α → Θ_{[λ,a,Ω,Γ_{+},α]} is increasing, we find from (3.31) that the point-wise limit (3.25) is well defined.

Let O_{1} ⊂ O two open subsets of Ω such that Ō_{1} ⊂ O, Ō ⊂ Ω, and choose δ > 0 sufficiently small so that Ō ⊂ D. Then, (3.31) implies

Θ_{[λ,a,Ω,Γ_{+},α]} ≤ Θ_{[λ,a,D,γ_{∞}^{D},M]} in O ,

for each α ∈ (0, ∞)^{q}, and hence, by the L^{p}-estimates of Agmon, Douglis & Nirenberg, for each p > 1 there exists a constant M_{1} = M(p, O_{1}) such that

∥Θ_{[λ,a,Ω,Γ_{+},α]}∥_{W^{2,p}(O_{1})} ≤ M_{1}

for each α ∈ (0, ∞)^{q}. Thus, thanks to Morrey’s embedding theorem and Schauder’s estimates, there exists a constant M_{2} = M(O_{1}) such that

∥Θ_{[λ,a,Ω,Γ_{+},α]}∥_{C^{2+μ}(Ō_{1})} ≤ M_{2} , α ∈ (0, ∞)^{q} .

Now, a rather standard compactness argument combined with the uniqueness of the point-wise limit (3.25) shows that Θ_{[λ,a,Ω,Γ_{+},∞]} ∈ C^{2+μ}(Ō_{1}) is a regular solution of

−Δu = λu − auf(·, u)

in O_{1}, and that

lim_{α↑∞} ∥Θ_{[λ,a,Ω,Γ_{+},α]} − Θ_{[λ,a,Ω,Γ_{+},∞]}∥_{C^{2+μ}(Ō_{1})} = 0 .  (3.32)

This completes the proof of Parts (i) and (ii).

We now prove Part (iii). By construction, for each α ∈ (0, ∞)^{q} we have that

Θ_{[λ,a,Ω,Γ_{+},α]} < Θ_{[λ,a,Ω,Γ_{+},∞]} .  (3.33)

Therefore, thanks to (3.2),

lim_{λ↑σ_{1}} Θ_{[λ,a,Ω,Γ_{+},∞]} = ∞ uniformly in ∪_{j=1}^{n_{1}} Ω̄_{0,j}^{1} \ ∂Ω .  (3.34)

We now show that the point-wise limit (3.28) is well defined. Reduce δ > 0, if necessary, so that the open neighborhoods

Ω_{0,j}^{1,δ} := {x ∈ Ω : dist(x, Ω_{0,j}^{1}) < δ} , 1 ≤ j ≤ n_{1} ,

satisfy

Ω̄_{0,j}^{1,δ} ∩ Ω̄_{0,ℓ}^{1,δ} = ∅

for any j ≠ ℓ, and consider the open set

D_{1} := D \ ∪_{j=1}^{n_{1}}Ω̄_{0,j}^{1,δ} .

Then,

∂D_{1} = ∂D ∪ ∪_{j=1}^{n_{1}}(∂Ω_{0,j}^{1,δ} \ ∂Ω) = γ_{0} ∪ γ_{∞}^{D} ∪ ∪_{j=1}^{n_{1}}(∂Ω_{0,j}^{1,δ} \ ∂Ω) .

By construction, the set

K_{1} := γ_{∞}^{D} ∪ ∪_{j=1}^{n_{1}}(∂Ω_{0,j}^{1,δ} \ ∂Ω)

is a compact subset of Ω_{+}. Thus, thanks to Theorem 3.3, there exists a universal constant M_{3} > 0 such that for any λ < σ_{1}

∥Θ_{[λ,a,Ω,Γ_{+},∞]}∥_{C(K_{1})} ≤ M_{3} .

Hence, for each λ < σ_{1} the function Θ_{[λ,a,Ω,Γ_{+},∞]} provides us with a subsolution of the problem

\begin{matrix}−Δu = λu − a(x)uf(x, u) in D_{1} , \\ u = M_{3} on K_{1} , \\ u = 0 on γ_{0} = ∂D_{1} \ K_{1} . \end{matrix}  (3.35)

The lower order refuges of D_{1} are Ω_{0,j}^{2}, 1 ≤ j ≤ n_{2}. Therefore, thanks to Theorem 3.1 the problem (3.35) possesses a regular positive solution if, and only if, λ < σ_{2}. Moreover, it is unique if it exists. Let Θ_{[σ_{1},a,D_{1},K_{1},M_{3}]} denote the unique regular positive solution of (3.35) for λ = σ_{1}. Thanks to Remark 3.2(b), for each λ < σ_{1} we have

Θ_{[λ,a,Ω,Γ_{+},∞]} ≤ Θ_{[σ_{1},a,D_{1},K_{1},M_{3}]} in D_{1} .  (3.36)

Since δ can be taken arbitrarily small and the mapping λ → Θ_{[λ,a,Ω,Γ_{+},∞]} is nondecreasing, it follows from (3.36) that the point-wise limit (3.28) is well defined. The same bootstrapping and compactness arguments as above combined with (3.28) show that Θ_{[σ_{1},a,Ω,Γ_{+},∞]} is a regular large solution of (3.29). Its minimal character follows easily from the construction itself. This completes the proof. □

4. Stabilization of the regular positive solutions in Ω \ Ω̄_{0}^{1} as λ ↑ σ_{1}.

Thanks to Theorem 2.1, the problem (1.5) possesses a regular positive solution if, and only if,

σ_{0} < λ < σ_{1} .

Moreover, it is unique if it exists. As in Section 2 we shall denote it by θ_{[λ,a,Ω]}. The following result shows that θ_{[λ,a,Ω]} converges to a large regular solution in Ω \ Ω̄_{0}^{1} as λ ↑ σ_{1}.

Theorem 4.1. Suppose (Ha1-3), (Hf) and (Hfb), and assume in addition that (3.7) is satisfied for any compact subset K of Ω_{+} and β = σ_{1}. Set

Ω_{1} := Ω \ Ω̄_{0}^{1} , Ω_{0}^{1} := ∪_{j=1}^{n_{1}}Ω_{0,j}^{1} .

Then, the point-wise limit

Ξ_{[σ_{1},a,Ω_{1}]}(x) := lim_{λ↑σ_{1}} θ_{[λ,a,Ω]}(x) x ∈ Ω_{1} ,  (4.1)

is well defined. Moreover, Ξ_{[σ_{1},a,Ω_{1}]} ∈ C^{2+μ}(Ω_{1}),

lim_{λ↑σ_{1}} ∥θ_{[λ,a,Ω]} − Ξ_{[σ_{1},a,Ω_{1}]}∥_{C^{2+μ}(Ω_{1})} = 0 ,  (4.2)

and Ξ_{[σ_{1},a,Ω_{1}]} is the minimal large regular solution of

−Δu = σ_{1}u − a(x)uf(x, u) in Ω_{1} ,
u = ∞ on ∂Ω_{1} ∩ Ω ,
u = 0 on ∂Ω_{1} ∩ ∂Ω ,

(4.3)

Proof. Basically this result is a particular case of Theorem 3.5(iii), but not exactly, since Theorem 3.5 dealt with the limiting behaviour of large regular solutions of (3.1) going to infinity on some of the components of ∂Ω_{+} ∩ ∂Ω and now it might occur ∂Ω_{+} ∩∂Ω = ∅. Nevertheless, the proof of Theorem 3.5(iii) carries over almost mutatis mutandis to prove Theorem 4.1. In the sequel the notations introduced in the proof of Theorem 3.5 will be kept out.

Pick up δ > 0 sufficiently small so that the open neighborhoods

Ω_{0,j}^{1,δ} := {x ∈ Ω : dist(x, Ω_{0,j}^{1}) < δ} , 1 ≤ j ≤ n_{1} ,

satisfy

Ω̄_{0,j}^{1,δ} ∩ Ω̄_{0,ℓ}^{1,δ} = ∅

for any j ≠ ℓ, and consider the open set

D := Ω \ ∪_{j=1}^{n_{1}}Ω̄_{0,j}^{1,δ} .

By definition, D ⊂ Ω_{1}. Moreover, any component of ∂Ω_{0,j}^{1,δ} \ ∂Ω, 1 ≤ j ≤ n_{1}, lies within Ω_{+}. Let Γ_{+}^{j}, 1 ≤ j ≤ q, denote all the components of ∂D contained in Ω_{+}. Note that any component of ∂D either it is entirely contained in Ω_{+}, or it is a component of ∂Ω. Then,

K := ∪_{j=1}^{q}Γ_{+}^{j}

is a compact subset of Ω_{+}. Thanks to Theorem 3.3, there exists a universal constant M > 0 such that

∥θ_{[λ,a,Ω]}∥_{C(K)} ≤ M

for any λ ∈ (σ_{0}, σ_{1}), and hence θ_{[λ,a,Ω]} provides us with a subsolution of

\begin{matrix}−Δu = λu − a(x)uf(x, u) in D , \\ u = M on K , \\ u = 0 on ∂D \ K . \end{matrix}  (4.4)

The lower order refuges of D are Ω_{0,j}^{2}, 1 ≤ j ≤ n_{2}. Therefore, thanks to Theorem 3.1 the problem (4.4) possesses a regular positive solution if, and only if, λ < σ_{2}. Moreover, it is unique if it exists. Let Θ_{[σ_{1},a,D,K,M]} denote the unique regular positive solution of (4.4) for λ = σ_{1}. Thanks to Remark 3.2(b) we have that

θ_{[λ,a,Ω]} ≤ Θ_{[σ_{1},a,D,K,M]} in D ,  (4.5)

for all λ ∈ (σ_{0}, σ_{1}). Since δ can be taken arbitrarily small,

lim_{δ↓0} D = Ω_{1} ,

and the mapping λ → θ_{[λ,a,Ω]} is increasing, it follows from (4.5) that the point-wise limit (4.1) is well defined.

Let O_{1} ⊂ O two open subsets of Ω_{1} such that Ō_{1} ⊂ O, Ō ⊂ Ω_{1}, and choose δ > 0 sufficiently small so that Ō ⊂ D. Then, (4.5) implies

θ_{[λ,a,Ω]} ≤ Θ_{[σ_{1},a,D,K,M]} in Ō , λ ∈ (σ_{0}, σ_{1}) .

Thus, by the L^{p}-estimates of Agmon, Douglis & Nirenberg, for each p > 1 there exists a constant C_{1} = C(p, O_{1}) such that

∥θ_{[λ,a,Ω]}∥_{W^{2,p}(O_{1})} ≤ C_{1} , λ ∈ (σ_{0}, σ_{1}) .

Hence, thanks to Morrey’s embedding theorem and Schauder’s estimates, there exists a constant C_{2} = C(O_{1}) such that

∥θ_{[λ,a,Ω]}∥_{C^{2+μ}(Ō_{1})} ≤ C_{2} , λ ∈ (σ_{0}, σ_{1}) .

Now, a well known compactness argument combined with the uniqueness of the point-wise limit (4.1) shows that Ξ_{[σ_{1},a,Ω_{1}]} ∈ C^{2+μ}(Ō_{1}) is a regular solution of

−Δu = σ_{1}u − auf(·, u)

in O_{1}, and that

lim_{λ↑σ_{1}} ∥θ_{[λ,a,Ω]} − Ξ_{[λ_{1},a,Ω_{1}]}∥_{C^{2+μ}(Ō_{1})} = 0 .

This completes the proof of (4.2). By (2.3) and the construction itself Ξ_{[σ_{1},a,Ω_{1}]} must be the minimal regular positive solution of (4.3). The proof is completed. □

5. The existence of metasolutions for λ ≥ σ_{1}.

The following concept goes back to [GL98] and [Go99], where it was introduced in the special case when f(x, u) = u^{p} for some p > 0.

Definition 5.1. Suppose (Ha1-3), (Hf) and set

Ω_{k} := Ω \ ∪_{i=1}^{k} ∪_{j=1}^{n_{i}} Ω̄_{0,j}^{i} , 1 ≤ k ≤ m .

A function

u : Ω → [0, ∞]

is said to be a regular metasolution of order k of (1.5) supported in Ω_{k} if u|_{Ω_{k}} is a large regular solution of

−Δu = λu − a(x)uf(x, u) in Ω_{k} ,
u = ∞ on ∂Ω_{k} ∩ Ω ,
u = 0 on ∂Ω_{k} ∩ ∂Ω ,

(5.1)

in the sense of the statement of Theorem 3.5(i), while

u = ∞ in (Ω \ Ω̄_{k}) ∪ (∂Ω_{k} ∩ Ω) .

Using this concept, the following result is an immediate consequence from Theorem 4.1.

Corollary 5.2. Suppose (Ha1-3), (Hf) and (Hfb), and assume in addition that (3.7) is satisfied for any compact subset K of Ω_{+} and β = σ_{1}. Then, the function M_{[σ_{1},a,Ω]} : Ω → [0, ∞] defined by

M_{[σ_{1},a,Ω]} := \{ \begin{matrix}Ξ_{[σ_{1},a,Ω_{1}]} & in Ω_{1} , \\ ∞ & in (Ω \ Ω̄_{1}) ∪ (∂Ω_{1} ∩ Ω) , \end{matrix}  (5.2)

is a regular metasolution of order one of (1.5), with λ = σ_{1}, supported in Ω_{1}.

The following result characterizes the range of values of the parameter λ for which (1.5) admits a regular metasolution of order k supported in Ω_{k}, 1 ≤ k ≤ m. Thanks to Theorem 2.2 those metasolutions are the candidates to describe the limiting behaviour of the population as time passes by for any λ ≥ σ_{1}.

Theorem 5.3. Suppose (Ha1-3), (Hf) and (Hfb), and assume in addition that (3.7) is satisfied for any compact subset K of Ω_{+} and β > 0. Fix k ∈ {1, …, m}. If k < m, then (1.5) possesses a regular metasolution of order k supported in Ω_{k} if, and only if, λ < σ_{k+1}. If k = m, then (1.5) possesses a regular metasolution of order m supported in Ω_{m} for each λ ∈ ℝ. Moreover, if the problem possesses a metasolution of order k supported in Ω_{k}, then it also possesses a minimal metasolution of order k supported in Ω_{k}, 1 ≤ k ≤ m.

Proof. Suppose k < m and (5.1) possesses a large solution u. Then,

λ ≤ σ^{Ω_{k}}[−Δ + af(·, u)] < σ^{Ω_{0,1}^{k+1}} = σ_{k+1} .

Suppose k < m and λ < σ_{k+1}. To prove the existence of a metasolution of order k supported in Ω_{k} it suffices to show that (5.1) has a large regular solution. As a(x) vanishes on ∂Ω_{k} \ ∂Ω, Theorem 3.5 can not be applied straight away to prove this, but this trouble may be overcome arguing as follows.

Fix M > 0 and consider the problem

−Δu = λu − a(x)uf(x, u) in Ω_{k} ,
u = M on ∂Ω_{k} ∩ Ω ,
u = 0 on ∂Ω_{k} ∩ ∂Ω .

(5.3)

This problem does not fit into the setting of Theorem 3.1, since a(x) = 0 on ∂Ω_{k} \ ∂Ω, but we can slightly modify Ω_{k} so that the corresponding problem does it. For each δ > 0 consider the open neighborhoods

Ω_{0,j}^{i,δ} := {x ∈ Ω : dist(x, Ω_{0,j}^{i}) < δ} , 1 ≤ i ≤ k , 1 ≤ j ≤ n_{i} .

By (Ha1-3) there exists δ_{0} > 0 such that for each δ ∈ (0, δ_{0})

Ω̄_{0,j}^{i,δ} ∩ Ω̄_{0,ĵ}^{î,δ} = ∅

if (i, j) ≠ (î, ĵ). Now, consider the open set

Ω_{k}^{δ} := Ω \ ∪_{i=1}^{k} ∪_{j=1}^{n_{i}} Ω̄_{0,j}^{i,δ} .

By definition,

Ω_{k}^{δ} ⊂ Ω \ ∪_{i=1}^{k} ∪_{j=1}^{n_{i}} Ω̄_{0,j}^{i} = Ω_{k} ,

and

lim_{δ↓0} Ω_{k}^{δ} = Ω_{k} ,

in the sense of [Lo96]. Moreover, any component of ∂Ω_{0,j}^{i,δ} \ ∂Ω, 1 ≤ i ≤ k, 1 ≤ j ≤ n_{i}, lies within Ω_{+}. Note that any component of ∂Ω_{k}^{δ} either it is entirely contained in Ω_{+}, or it is a component of ∂Ω. Now, consider the modified problem

−Δu = λu − a(x)uf(x, u) in Ω_{k}^{δ} ,
u = M on ∂Ω_{k}^{δ} ∩ Ω ,
u = 0 on ∂Ω_{k}^{δ} ∩ ∂Ω .

(5.4)

Thanks to Theorem 3.1, (5.4) possesses a unique positive solution, say Θ_{[λ,a,δ,M]}. Let O_{1} ⊂ O be two open subdomains of Ω_{k} such that Ō_{1} ⊂ O, Ō ⊂ Ω_{k}. By construction, there exists δ_{1} ∈ (0, δ_{0}) such that for any δ ∈ (0, δ_{1})

Ō ⊂ Ω_{k}^{δ_{1}} ⊂ Ω_{k}^{δ} ⊂ Ω_{k} ,

Set

K := ∂Ω_{k}^{δ_{1}} \ ∂Ω .

By construction, for each δ ∈ (0, δ_{1}) K is a compact subset of

{x ∈ Ω_{k}^{δ} : a(x) > 0} .

Thus, thanks to Theorem 3.3, there exists a constant M_{1} > 0 such that for each δ ∈ (0, δ_{1})

∥Θ_{[λ,a,δ,M]}∥_{C(K)} ≤ M_{1} .

Hence, for each 0 < δ < δ_{1} the function Θ_{[λ,a,δ,M]} provides us with a positive subsolution of

−Δu = λu − a(x)uf(x, u) in Ω_{k}^{δ_{1}} ,
u = M_{1} on ∂Ω_{k}^{δ_{1}} ∩ Ω ,
u = 0 on ∂Ω_{k}^{δ_{1}} ∩ ∂Ω .

(5.5)

Let Θ_{[λ,a,δ_{1},M_{1}]} denote the unique positive solution of (5.5), whose existence is guaranteed by Theorem 3.1. Thanks to Remark 3.2(b), for each δ ∈ (0, δ_{1}) we have

Θ_{[λ,a,δ,M]} ≤ Θ_{[λ,a,δ_{1},M_{1}]} in Ω_{k}^{δ_{1}} ,

and therefore, there exists a constant M_{2} > 0 such that for each 0 < δ < δ_{1}

∥Θ_{[λ,a,δ,M]}∥_{C(Ō)} ≤ M_{2} .

By the same bootstrapping argument used in the proof of Theorem 3.5 and Theorem 4.1, there exists a constant M_{3} > 0 such that for each δ ∈ (0, δ_{1})

∥Θ_{[λ,a,δ,M]}∥_{C^{2+μ}(Ō_{1})} ≤ M_{3} .

Now, by a standard compactness argument combined with a diagonal procedure it is clear that we can substract a subsequence δ_{n} ↓ 0 such that

lim_{n→∞} ∥Θ_{[λ,a,δ_{n},M]} − Θ_{[λ,a,M]}∥_{C^{2+μ}(Ω_{k})} = 0

for some Θ_{[λ,a,M]} ∈ C^{2+μ}(Ω_{k}). Necessarily,

lim_{x→[∂Ω_{k}\∂Ω]} Θ_{[λ,a,M]}(x) = M .

Therefore, Θ_{[λ,a,M]} provides us with a solution of (5.3). The same argument of the proof of Theorem 3.1 shows that in fact Θ_{[λ,a,M]} is the unique regular positive solution of (5.3), and that M → Θ_{[λ,a,M]} is point-wise increasing.

Thanks to Theorem 3.3, there exists a constant M_{4} > 0 such that for any M > 0

∥Θ_{[λ,a,M]}∥_{C(K)} ≤ M_{4} .

Hence, for each M > 0 the function Θ_{[λ,a,M]} provides us with a subsolution of

−Δu = λu − a(x)uf(x, u) in Ω_{k}^{δ_{1}} ,
u = M_{4} on ∂Ω_{k}^{δ_{1}} ∩ Ω ,
u = 0 on ∂Ω_{k}^{δ_{1}} ∩ ∂Ω ,

(5.6)

and so

Θ_{[λ,a,M]} ≤ Θ_{[λ,a,δ_{1},M_{4}]} in Ω_{k}^{δ_{1}} .

Therefore, the point-wise limit

Θ_{[λ,a,∞]}(x) := lim_{M↑∞} Θ_{[λ,a,M]}(x) , x ∈ Ω_{k} ,

is well defined. Finally, the same regularity and compactness argument given before shows that Θ_{[λ,a,∞]} is a large solution of (5.1). Its minimality follows easily from the construction itself.

In case k = m, by Remark 3.2(a) we do not have any limitation on the size of λ in order to apply Theorem 3.1, and therefore the previous procedure provides us with a minimal large solution of (5.1) for each λ ∈ ℝ. This completes the proof. □

The following result shows the point-wise behavior of the metasolutions of order k ≤ m − 1 supported in Ω_{k} as λ ↑ σ_{k+1}. They stabilize in Ω_{k+1}, whereas they grow to infinity in

∪_{j=1}^{n_{k+1}} Ω̄_{0,j}^{k+1} \ ∂Ω .

Therefore, they provide us with metasolutions of order k + 1 supported in Ω_{k+1}.

Theorem 5.4. Suppose (Ha1-3), (Hf) and (Hfb), and assume in addition that (3.7) is satisfied for any compact subset K of Ω_{+} and β > 0. Fix k ∈ {1, …, m − 1} and consider a sequence

λ_{n} ∈ (σ_{k}, σ_{k+1}) , n ≥ 1 ,

such that

lim_{n→∞} λ_{n} = σ_{k+1} .

For each n ≥ 1, let u_{n} be a metasolution of order k of (1.5), with λ = λ_{n}, supported in

Ω_{k} = Ω \ ∪_{i=1}^{k} ∪_{j=1}^{n_{i}} Ω̄_{0,j}^{i} ,

whose existence is guaranteed by Theorem 5.3. Then,

lim_{n→∞} u_{n} = ∞ uniformly in ∪_{j=1}^{n_{k+1}} Ω̄_{0,j}^{k+1} \ ∂Ω .  (5.7)

Moreover, for any open subdomain O of Ω_{k+1} with Ō ⊂ Ω_{k+1} there exists a subsequence of (λ_{n}, u_{n}), n ≥ 1, relabeled again by n, and a regular solution of

−Δu = σ_{k+1}u − auf(·, u)

in O, say u_{ω} ∈ C^{2+μ}(Ō), such that

lim_{n→∞} ∥u_{n} − u_{ω}∥_{C^{2+μ}(Ō)} = 0 .  (5.8)

Furthermore, if u_{λ} stands for the minimal metasolution of order k of (1.5) supported in Ω_{k}, then the point-wise limit

u_{ω}(x) = lim_{λ↑σ_{k+1}} u_{λ}(x) , x ∈ Ω_{k+1} ,

is well defined and its extension by ∞ to Ω provides us with a metasolution of order k + 1 of (1.5) supported in Ω_{k+1} (for the value of the parameter λ = σ_{k+1}).

Proof. By definition, for each n ≥ 1 the function u_{n}|_{Ω_{k}} is a large regular solution of

−Δu = λ_{n}u − a(x)uf(x, u) in Ω_{k} ,
u = ∞ on ∂Ω_{k} ∩ Ω ,
u = 0 on ∂Ω_{k} ∩ ∂Ω .

In particular, u_{n}|_{Ω_{k}} is a positive strict supersolution of

\begin{matrix}−Δu = λ_{n}u − a(x)u^{p+1} in Ω_{k} , \\ u = 0 on ∂Ω_{k} , \end{matrix}  (5.9)

and hence,

u_{n}|_{Ω_{k}} ≥ θ_{[λ_{n},a,Ω_{k}]} ,  (5.10)

where θ_{[λ_{n},a,Ω_{k}]} ≫ 0 is the unique positive solution of (5.9), whose existence and uniqueness is guaranteed by Theorem 2.1(i). Note that the lower order refuges of Ω_{k} are Ω_{0,j}^{k+1}, 1 ≤ j ≤ n_{k+1}, and that

λ_{n} < σ_{k+1} = σ^{Ω_{0,j}^{k+1}} .

Thanks to Theorem 2.1(iii), we have

lim_{n→∞} θ_{[λ_{n},a,Ω_{k}]} = ∞ uniformly in ∪_{j=1}^{n_{k+1}} Ω̄_{0,j}^{k+1} \ ∂Ω_{k} ,

and hence we find from (5.10) that

lim_{n→∞} u_{n} = ∞ uniformly in ∪_{j=1}^{n_{k+1}} Ω̄_{0,j}^{k+1} \ ∂Ω_{k} .  (5.11)

Moreover, u_{n} = ∞ in (Ω \ Ω̄_{k}) ∪ (∂Ω_{k} ∩ Ω), since u_{n} is a metasolution of order k supported in Ω_{k}. Thus, (5.11) implies (5.7).

Now, for each δ > 0 sufficiently small consider the open set

D := Ω \ ∪_{i=1}^{k+1} ∪_{j=1}^{n_{i}} Ω̄_{0,j}^{i,δ} ,

where

Ω_{0,j}^{1,δ} := {x ∈ Ω : dist(x, Ω_{0,j}^{i}) < δ} , 1 ≤ i ≤ k + 1 , 1 ≤ j ≤ n_{i} .

By definition,

D ⊂ Ω \ ∪_{i=1}^{k+1} ∪_{j=1}^{n_{i}} Ω̄_{0,j}^{i} = Ω_{k+1} ,

and

lim_{δ↓0} D = Ω_{k+1} ,

in the sense of [Lo96]. Moreover, any component of ∂Ω_{0,j}^{i,δ} \ ∂Ω, 1 ≤ i ≤ k + 1, 1 ≤ j ≤ n_{i}, lies within Ω_{+}. Let Γ_{+}^{j}, 1 ≤ j ≤ q, denote all the components of ∂D contained in Ω_{+}. Note that any component of ∂D either it is entirely contained in Ω_{+}, or it is a component of ∂Ω. Then,

K := ∪_{j=1}^{q}Γ_{+}^{j}

is a compact subset of Ω_{+}. Thanks to Theorem 3.3, there exists a constant M > 0 such that for any n ≥ 1,

∥u_{n}∥_{C(K)} ≤ M ,

and hence u_{n} provides us with a positive subsolution of

\begin{matrix}−Δu = σ_{k+1}u − a(x)uf(x, u) in D , \\ u = M on K , \\ u = 0 on ∂D \ K . \end{matrix}  (5.12)

The lower order refuges of (5.12) are Ω_{0,j}^{k+2}, 1 ≤ j ≤ n_{k+2}, if k ≤ m − 2, and a(x) is bounded away from zero if k = m − 1. Moreover, σ_{k+1} < σ_{k+2}. Therefore, thanks to Theorem 3.1 the problem (5.12) possesses a unique regular positive solution. Let Θ denote it. By Remark 3.2(b) we find that for each n ≥ 1

u_{n} ≤ Θ in D .  (5.13)

Let O be an open subset of Ω_{k+1} with Ō ⊂ Ω_{k+1} and choose δ > 0 sufficiently small so that

O ⊂ D .

The same bootstrapping argument of the proof of Theorem 4.1 shows that there exists a constant M_{1} > 0 such that

∥u_{n}∥_{C^{2+μ}(Ō)} ≤ M_{1} , n ≥ 1 .

Therefore, we can substract a subsequence of (λ_{n}, u_{n}), again labeled by n, such that

lim_{n→∞} ∥u_{n} − u_{ω}∥_{C^{2+μ}(Ō)} = 0

for some u_{ω} ∈ C^{2+μ}(Ō). Necessarily, u_{ω} is a solution of

−Δu = σ_{k+1}u − auf(·, u)

in O.

The last assertion follows easily from the fact that λ → u_{λ}(x) is nondecreasing for each x ∈ Ω̄. This completes the proof of the theorem. □

6. The asymptotic behaviour of the population for λ ≥ σ_{1}.

In this section we characterize the asymptotic behaviour of the population as t ↑ ∞ for any value of the parameter λ ≥ σ_{1}.

Theorem 6.1. Suppose (Ha1-3), (Hf) and

∂Ω ∩ ∂Ω_{+} ≠ ∅ .

Let Γ_{+}^{j}, 1 ≤ j ≤ q, be q arbitrary components of ∂Ω ∩ ∂Ω_{+}, and α_{j} > 0, 1 ≤ j ≤ q, q arbitrary constants, and consider the evolution problem

\frac{∂u}{∂t} − Δu = λu − a(x)uf(x, u) in Ω × (0, ∞) ,

u|_{Γ_{+}^{j}} = α_{j} > 0 , 1 ≤ j ≤ q t > 0
u = 0 on (∂Ω \ ∪_{j=1}^{q}Γ_{+}^{j}) × (0, ∞) ,
u(·, 0) = u_{0} , in Ω ,

(6.1)

The following assertions are true:

(i) For each u_{0} ∈ U_{0}, u_{0} ≥ 0, the problem (6.1) possesses a unique global regular solution u_{[λ,a,Ω,Γ_{+},α]}(·, ·; u_{0}) ∈ C^{2+μ,1+ \frac{μ}{2}} (Ω̄ × (0, ∞)).

(ii) For each u_{0} ∈ U_{0}, u_{0} ≥ 0,

lim_{t→∞} ∥u_{[λ,a,Ω,Γ_{+},α]}(·, t; u_{0}) − Θ_{[λ,a,Ω,Γ_{+},α]}∥_{C(Ω̄)} = 0 ,  (6.2)

where Θ_{[λ,a,Ω,Γ_{+},α]} is the unique regular positive solution of (3.1).

Proof. The existence of a unique regular solution follows from the results of [DK92]. The global existence of these solutions is easily obtained from the estimate

\frac{∂u}{∂t} − Δu = λu − auf(·, u) ≤ λu .

This completes the proof of Part (i). Adapting the proof of Theorem 2.2 it is easily seen that for any u_{0} ∈ U_{0}, u_{0} ≥ 0, condition (6.2) holds. □

From this result it easily follows the main theorem of this section.

Theorem 6.2. Suppose (Ha1-3), (Hf) and (Hfb), and assume in addition that (3.7) is satisfied for any compact subset K of Ω_{+} and β > 0 and that either 1 ≤ k ≤ m−1 and σ_{k} ≤ λ < σ_{k+1}, or k = m and λ ≥ σ_{m}. Then, for any u_{0} ∈ U_{0} there exists ν > 0 such that for any compact subset K of Ω_{k} the restriction of the orbit

Γ(u_{0}) := {u_{[λ,a,Ω]}(·, t; u_{0}) : t ≥ 0}

to K is relatively compact in C^{2+ν}(K). Moreover,

Θmin_{[λ,a,∞]} ≤ lim inf_{t↑∞} u_{[λ,a,Ω]}(·, t; u_{0}) ≤ lim sup_{t↑∞} u_{[λ,a,Ω]}(·, t; u_{0}) ≤ Θmax_{[λ,a,Ω]} ,

where Θmin_{[λ,a,∞]} and Θmax_{[λ,a,∞]} stand for the minimal and maximal large solutions of problem (5.1), respectively. Thus, if (5.1) possesses a unique large solution, say Θ_{[λ,a,Ω]}, then

lim_{t↑∞} ∥u_{[λ,a,Ω]}(·, t; u_{0}) − Θ_{[λ,a,Ω]}∥_{C^{2+ν}(K)} = 0  (6.3)

for any compact subset K of Ω_{k}, and therefore due to Theorem 2.2(iii) the solution u_{[λ,a,Ω]}(·, t; u_{0}) is point-wise convergent as t ↑ ∞ to the metasolution of order k supported in Ω_{k} associated with Θ_{[λ,a,Ω]}.

Proof. The notations introduced in the proof of Theorem 5.3 will be kept out. Consider 1 ≤ k ≤ m and λ within its corresponding range of values. Let K ⊂ Ω_{k} compact and consider δ > 0 sufficiently small so that

K ⊂ Ω_{k}^{δ} ⊂ Ω_{k} .

Set

M_{L} := inf_{∂Ω_{k}^{δ}∩Ω} u_{[λ,a,Ω]}(·, 1; u_{0}) , M_{S} := sup_{∂Ω_{k}^{δ}∩Ω} u_{[λ,a,Ω]}(·, 1; u_{0})

and consider the auxiliary evolution problems

\frac{∂u}{∂t} − Δu = λu − a(x)uf(x, u) in Ω_{k}^{δ} × (0, ∞) ,

u = M > 0 , on (∂Ω_{k}^{δ} ∩ Ω) × (0, ∞) ,
u = 0 on (∂Ω_{k}^{δ} ∩ ∂Ω) × (0, ∞) ,
u(·, 0) = u_{1} , in Ω̄_{k}^{δ} ,

(6.4)

where

M ∈ {M_{L}, M_{s}} , u_{1} := u_{[λ,a,Ω]}(·, 1; u_{0}) .

Thanks to the parabolic maximum principle, for each t ≥ 1 we have that

u_{[λ,a,δ,M_{L}]}(·, t − 1; u_{1}) ≤ u_{[λ,a,Ω]}(·, t; u_{0}) ≤ u_{[λ,a,δ,M_{S}]}(·, t − 1; u_{1}) in Ω_{k}^{δ} ,

where u_{[λ,a,δ,M]}(·, t − 1; u_{1}) is the unique global regular solution of (6.4), whose existence is guaranteed by Theorem 6.1(i). Moreover, it follows from Theorem 6.1(ii) that

lim_{t↑∞} ∥u_{[λ,a,δ,M]}(·, t − 1; u_{1}) − Θ_{[λ,a,δ,M]}∥_{C^{1}(Ω̄_{k}^{δ})} = 0 ,  (6.5)

where Θ_{[λ,a,δ,M]} is the unique regular positive steady-state of (6.4), whose existence is guaranteed by Theorem 3.1. In particular, for each δ > 0 we have that

Θ_{[λ,a,δ,M_{L}]} ≤ lim inf_{t↑∞} u_{[λ,a,Ω]}(·, t; u_{0}) ≤ lim sup_{t↑∞} u_{[λ,a,Ω]}(·, t; u_{0}) ≤ Θ_{[λ,a,δ,M_{S}]} (6.6)

in Ω̄_{k}^{δ}. On the other hand, thanks to Theorem 2.2(iii) and the parabolic estimates of [Re82] and [Re86] it is easily seen that

lim_{δ↓0} M_{L} = ∞ .

Thus, thanks to the analysis already done in the proof of Theorem 5.3,

Θmin_{[λ,a,Ω]} ≤lim inf_{t↑∞} u_{[λ,a,Ω]}(·, t; u_{0})≤lim sup_{t↑∞} u_{[λ,a,Ω]}(·, t; u_{0})≤ inf_{δ>0} Θ_{[λ,a,δ,∞]} =Θmax_{[λ,a,Ω]} .

This completes the proof of (6.3). Moreover, it shows that there exists a constant M > 0 such that

∥u_{[λ,a,Ω]}(·, t; u_{0})∥_{C(K)} ≤ M .

Hence, by the results of [Re86], the restriction of Γ(u_{0}) to K, say Γ_{K}(u_{0}), is relatively compact in C^{2−}(K). Therefore, by the parabolic Schauder estimates, [LSU68], there exists ν > 0 such that Γ_{K}(u_{0}) is relatively compact in C^{2+ν}(K). This completes the proof. □

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Julián López-Gómez Departamento de Matemática Aplicada Universidad Complutense de Madrid 28040-MADRID, Spain. Phone (34) 91-394-4194, Fax (34) 91-394-4102

   E-mail address: Lopez Gomez@mat.ucm.es